#6536572

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1925 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)(3,6)(4,5)(7,8)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,32)(28,31);(9,10)(11,14)(12,13)(15,16)(17,19,20,28,22,27,21,29)(18,23,30,32,25,31,24,26);(2,5)(6,8)(10,13)(14,16)(17,18)(19,23)(20,30)(21,24)(22,25)(26,29)(27,31)(28,32);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(19,27)(23,31)(26,32)(28,29);(11,15)(14,16)(17,20,22,21)(18,24,25,30)(19,28,27,29)(23,26,31,32)

Polycyclic presentation:

2^9|b^2=fi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^c=ei,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257398, #72257577, #72257627
maximal normal subgroup of	#72257398, #72257577, #72257627
maximal quotient of	#72265547

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6443768);