#6399835

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	C₂×#547	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2181 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)(15,20)(16,28)(17,23)(18,21)(19,24)(22,30)(25,39)(26,33)(27,36)(29,38)(31,35)(32,37)(34,42)(40,43)(41,44);(6,9)(7,11)(13,15,19,27)(14,20,24,36)(16,25,30,42)(17,31,18,26)(21,33,23,35)(22,34,28,39)(29,41,32,40)(37,43,38,44);(14,21)(17,18)(20,33)(22,37)(23,24)(26,31)(28,38)(29,32)(34,43)(35,36)(39,44)(40,41);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,25)(17,32)(18,29)(19,30)(20,34)(21,37)(23,38)(24,28)(26,41)(27,42)(31,40)(33,43)(35,44)(36,39);(1,3,4,2)(6,9)(10,12)(13,17,19,18)(14,23,24,21)(15,26,27,31)(16,29,30,32)(20,33,36,35)(22,37,28,38)(25,40,42,41)(34,44,39,43)

Polycyclic presentation:

2^9|a^2=h,b^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,e^b=ei,e^c=ei,g^c=gi,e^d=ei

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, 6307031);