#6459688

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3013	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3063 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366884);