#7052070

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	#45353	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1614 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)(17,18,23,27)(19,24,22,26)(20,29,31,28)(21,25,32,30);(1,2)(3,7)(4,8)(5,6)(10,11)(12,13)(17,19,23,22)(18,24,27,26)(20,30,31,25)(21,28,32,29);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,15,13,16)(17,20,22,30,23,31,19,25)(18,21,24,29,27,32,26,28);(1,3,5,4)(2,8,6,7)(9,12,14,13)(10,15,11,16)(17,21,23,32)(18,25,27,30)(19,28,22,29)(20,26,31,24);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,15,11,16)(17,19,23,22)(18,26,27,24)(20,30,31,25)(21,29,32,28)

Polycyclic presentation:

2^9|a^2=h,b^2=i,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=ehi,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, 6959266);