#7052126

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#45417	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1515 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6959322);