#7608244

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

1353 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)(17,18,23,25)(19,20,28,26)(21,22,31,32)(24,29,30,27);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,19,22,27,23,28,32,29)(18,24,21,20,25,30,31,26);(1,4,5,2)(3,7,8,6)(10,15)(11,16)(17,20)(18,19)(21,27)(22,30)(23,26)(24,32)(25,28)(29,31);(3,8)(6,7)(9,12,14,13)(10,11,15,16)(18,25)(20,26)(21,31)(24,30);(9,13,14,12)(10,11,15,16)(17,21,22,25,23,31,32,18)(19,26,27,24,28,20,29,30)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=ghi,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^c=ei,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, 7515440);