#346260

order	512
nilpotency class	3
isoclinism	isoclinic class of #346224
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

544 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,11,12,7)(4,13)(5,15,16,9)(6,10)(8,14)(17,18)(19,27,30,23)(20,31,34,24)(21,25)(22,26)(28,41)(29,37,43,39)(32,36)(33,44,46,38)(35,40)(42,47)(45,48);(1,3,6,12)(2,7,10,11)(4,9,14,15)(5,8,16,13)(17,19)(18,23)(20,32)(21,29)(22,30)(24,36)(25,39)(26,27)(28,34)(31,41)(33,45)(35,43)(37,40)(38,48)(42,46)(44,47);(1,4)(2,8,10,13)(3,9,12,15)(5,11)(6,14)(7,16)(17,20)(18,24)(19,28,30,32)(21,33)(22,34)(23,36,27,41)(25,38)(26,31)(29,42,43,45)(35,46)(37,47,39,48)(40,44);(1,5,6,16)(2,9,10,15)(3,8,12,13)(4,11,14,7)(17,21,22,35)(18,25,26,40)(19,29,30,43)(20,33,34,46)(23,37,27,39)(24,38,31,44)(28,42,32,45)(36,47,41,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=ef,d^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=de,f^b=fh,e^c=ei,f^c=fi,g^c=gh,g^d=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283654, #72283815, #72283871, …
maximal normal subgroup of	#72283654, #72283815, #72283871, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253456);