#374511

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=efh,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281707);