#7590760

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1295 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7497956);