#6811062

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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2100 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)(5,6,9,11)(7,12,8,10)(13,14,18,19)(15,17,22,27)(16,23,25,20)(21,26,28,24);(5,6)(7,10)(8,12)(9,11)(13,15,18,22)(14,17,19,27)(16,24,25,26)(20,28,23,21);(2,4)(13,16,19,23,18,25,14,20)(15,21,17,24,22,28,27,26);(5,7,9,8)(6,10,11,12)(16,25)(20,23)(21,28)(24,26);(5,8,9,7)(6,10,11,12)(13,17,19,22,18,27,14,15)(16,26,23,21,25,24,20,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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.

Relationships

maximal subgroup of	#72308311, #72308384
maximal normal subgroup of	#72308311, #72308384

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718258);