#6814925

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1946 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)(13,14,19,24)(15,18,23,20)(16,21,26,28)(17,25,27,22);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,27)(17,26)(18,24)(19,23)(21,25)(22,28);(1,2,3,4)(13,16,27,15,19,26,17,23)(14,21,22,18,24,28,25,20);(6,10)(9,11)(13,17,19,27)(14,22,24,25)(15,26,23,16)(18,21,20,28);(7,12)(9,11)(13,18,17,21,19,20,27,28)(14,23,25,26,24,15,22,16)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,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,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722121);