#7199167

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1934 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)(7,10)(8,11)(9,12)(13,14,17,16,18,20,25,26)(15,21,24,28,23,19,27,22);(5,6)(7,10)(8,11)(9,12)(13,14)(15,21)(16,25)(17,26)(18,20)(19,23)(22,24)(27,28);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,15,18,23)(14,19,20,21)(16,22,26,28)(17,24,25,27);(5,8,10,9)(6,11,7,12)(13,16,25,14,18,26,17,20)(15,22,27,19,23,28,24,21);(5,9,10,8)(6,11,7,12)(14,20)(15,23)(16,26)(24,27)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=fh,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,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, 7106363);