#7051953

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	#45023	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1573 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,c^2=fgi,d^2=fi,e^2=f,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=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, 6959149);