#7108107

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993979
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	#39220	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1412 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)(17,18,21,25)(19,22,26,30)(20,23,28,24)(27,29,32,31);(1,2)(3,8)(4,7)(5,6)(10,11)(12,13)(17,19,21,26)(18,22,25,30)(20,27,28,32)(23,29,24,31);(9,11,14,10)(12,15,13,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,15,11,16);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,15,11,16)(19,26)(22,30)(27,32)(29,31)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=f,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=ehi,d^c=dh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7015303);