#6506718

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,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, 6413914);