#7135118

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#44936	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1719 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=h,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7042314);