#346248

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

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

normal subgroups only

This group has 551 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)(3,9,11,6)(4,12,14,7)(5,8)(10,16)(13,15)(17,18)(19,23)(20,24,31,39)(21,32,33,25)(22,26)(27,34,41,37)(28,42,43,35)(29,36)(30,44)(38,46)(40,48)(45,47);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,15,12,16)(17,19,22,29)(18,23,26,36)(20,27,31,41)(21,28,33,43)(24,37,39,34)(25,35,32,42)(30,45,46,40)(38,47,44,48);(1,4)(2,7)(3,10,11,13)(5,14)(6,15,9,16)(8,12)(17,20)(18,24,26,39)(19,27)(21,30)(22,31)(23,34,36,37)(25,38,32,44)(28,40)(29,41)(33,46)(35,47,42,48)(43,45);(6,9)(15,16)(17,21)(18,25)(19,28)(20,30)(22,33)(23,35)(24,38)(26,32)(27,40)(29,43)(31,46)(34,47)(36,42)(37,48)(39,44)(41,45)

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,c^2=ef,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	#72283491, #72283593, #72283654, …
maximal normal subgroup of	#72283491, #72283593, #72283654, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253444);