#7608168

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	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1360 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=gh,d^2=gi,e^2=fgi,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, 7515364);