#373929

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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 ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1118 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,e^d=ei

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, 281125);