#6810817

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1881 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308367
maximal normal subgroup of	#72308367

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718013);