#6399497

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,g^c=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6306693);