#7149302

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

2040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7056498);