#367715

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

975 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egh,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,h^b=hi,g^c=gi,e^d=ei,h^d=hi

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, 274911);