#6399735

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2161 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)(15,27)(16,23)(17,33)(18,26)(19,25)(20,24)(21,30)(22,32)(28,37)(29,38)(31,39)(34,41)(35,40)(36,44)(42,43);(6,9)(7,11)(13,15)(14,21)(16,28)(17,34)(18,29)(19,30)(20,31)(22,36)(23,41)(24,38)(25,27)(26,39)(32,43)(33,37)(35,44)(40,42);(13,16,19,17)(14,22,25,40)(15,28,30,34)(18,32,20,35)(21,36,27,42)(23,26,33,24)(29,43,31,44)(37,38,41,39);(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,23,25,33)(15,28,30,34)(18,32,20,35)(21,37,27,41)(22,26,40,24)(29,44,31,43)(36,38,42,39);(1,3,4,2)(6,9)(10,12)(13,18,19,20)(14,24,25,26)(15,29,30,31)(16,32,17,35)(21,38,27,39)(22,23,40,33)(28,43,34,44)(36,41,42,37)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6306931);