#6814418

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,19)(15,20,24,28)(16,23,25,21)(17,22,26,27);(5,7)(6,9)(8,12)(10,11)(13,15,18,24)(14,20,19,28)(16,17,25,26)(21,27,23,22);(1,2,3,4)(13,16,14,21,18,25,19,23)(15,22,28,26,24,27,20,17);(6,10)(9,11)(13,14,18,19)(15,20,24,28)(16,23,25,21)(17,22,26,27);(7,12)(9,11)(13,17,19,27,18,26,14,22)(15,23,20,25,24,21,28,16)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6721614);