#7201607

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	#36009	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1709 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,26,30,32)(20,24,21,31)(23,28,27,29);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,18,22,25)(19,27,30,23)(20,31,21,24)(26,29,32,28);(11,15)(13,16)(17,19)(18,23)(20,29)(21,28)(22,30)(24,32)(25,27)(26,31);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,18)(19,23)(20,31)(21,24)(22,25)(26,29)(27,30)(28,32);(2,5)(6,8)(11,15)(13,16)(17,20,22,21)(18,24,25,31)(19,28,30,29)(23,26,27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,e^c=ei,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, 7108803);