#7135036

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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)(17,18,22,24,20,26,21,27)(19,25,30,32,28,23,29,31);(1,3)(2,6)(4,7)(5,8)(10,11)(12,13)(17,19,20,28)(18,23,26,25)(21,29,22,30)(24,31,27,32);(9,11,14,10)(12,15,13,16)(18,24)(21,22)(23,31)(25,32)(26,27)(29,30);(3,8)(6,7)(9,12,14,13)(10,15,11,16)(17,19)(18,25)(20,28)(21,29)(22,30)(23,26)(24,32)(27,31);(1,4,5,2)(3,7,8,6)(9,13,14,12)(10,15,11,16)(19,28)(23,25)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

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, 7042232);