#7230502

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#30729	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1489 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7137698);