#7117806

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,c^2=gh,d^2=fi,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 7025002);