#7195287

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	#44881	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1787 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)(10,13)(11,15)(17,18,22,25)(19,26,29,23)(20,28,30,31)(21,24,32,27);(1,3)(2,6)(4,7)(5,8)(10,13)(11,15)(17,19,22,29)(18,23,25,26)(20,24,30,27)(21,31,32,28);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20,19,27,22,30,29,24)(18,21,23,28,25,32,26,31);(3,8)(6,7)(10,13)(14,16)(17,21)(18,24)(19,28)(20,23)(22,32)(25,27)(26,30)(29,31);(9,11,12,15)(10,14,13,16)(18,25)(21,32)(23,26)(28,31)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgh,e^2=hi,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, 7102483);