#7206158

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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	#30499	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1688 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)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,32)(21,26)(22,24)(23,29)(25,31)(27,30);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,29)(18,23,26,28)(20,22,31,30)(24,32,27,25);(11,15)(13,16)(18,24)(19,29)(22,30)(23,25)(26,27)(28,32);(1,4,6,5)(2,7,3,8)(10,14)(11,15)(17,20,21,31)(18,25,26,32)(19,30,29,22)(23,27,28,24);(1,5,6,4)(2,7,3,8)(11,15)(13,16)(18,26)(19,29)(20,31)(24,27)

Polycyclic presentation:

2^9|b^2=f,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7113354);