#7007313

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1530 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,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, 6914509);