#346259

order	512
nilpotency class	3
isoclinism	isoclinic class of #346224
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

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

normal subgroups only

This group has 555 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)(4,12,13,7)(5,8)(10,15)(14,16)(17,18)(19,23)(20,24,32,39)(21,33,34,25)(22,26)(27,35,30,43)(28,41,42,36)(29,37)(31,45)(38,46)(40,48)(44,47);(1,3)(2,6)(4,10)(5,11)(7,15)(8,9)(12,14)(13,16)(17,19,22,29)(18,23,26,37)(20,30,32,27)(21,28,34,42)(24,35,39,43)(25,36,33,41)(31,40,46,44)(38,48,45,47);(1,4,5,13)(2,7,8,12)(3,10)(6,14)(9,15)(11,16)(17,20)(18,24,26,39)(19,27)(21,31)(22,32)(23,35,37,43)(25,38,33,45)(28,40)(29,30)(34,46)(36,47,41,48)(42,44);(6,9)(14,15)(17,21)(18,25)(19,28)(20,31)(22,34)(23,36)(24,38)(26,33)(27,40)(29,42)(30,44)(32,46)(35,47)(37,41)(39,45)(43,48)

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=efh,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	#72283721, #72283729, #72283847, …
maximal normal subgroup of	#72283721, #72283729, #72283847, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253455);