#7195278

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	#45316	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1631 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgh,e^2=h,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=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7102474);