#7044087

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	#45353	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1618 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=fgi,d^2=f,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, 6951283);