#7018451

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1365 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,32,22,26,21,27)(19,20,29,31,24,25,28,30);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,24)(18,20,26,25)(21,28,23,29)(27,30,32,31);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20,22,25)(18,24,26,19)(21,31,23,30)(27,28,32,29);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,19)(18,25)(20,26)(21,28)(22,24)(23,29)(27,31)(30,32);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(18,26)(19,24)(27,32)(28,29)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=fi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6925647);