#6815028

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2048 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)(6,9)(7,11)(13,14,18,22)(15,19,24,25)(16,23,27,20)(17,21,26,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,19,22,25)(16,26,27,17)(20,21,23,28);(1,2,3,4)(13,16,14,20,18,27,22,23)(15,21,25,26,24,28,19,17);(5,7,8,11)(6,10,9,12)(13,14,18,22)(15,19,24,25)(16,20,27,23)(17,28,26,21);(6,9)(10,12)(13,17,22,28,18,26,14,21)(15,23,19,27,24,20,25,16)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=fi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=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, 6722224);