#6534100

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#23393	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1869 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)(4,5)(7,8)(17,18)(19,27)(20,25)(21,26)(22,24)(23,29)(28,32)(30,31);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,30)(21,29)(22,28)(24,31)(25,32)(26,27);(10,13)(14,16)(18,24)(20,22)(23,31)(25,26)(27,32)(28,30);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,22)(18,25,26,24)(19,28,29,30)(23,31,27,32);(11,15)(14,16)(17,20,21,22)(18,25,26,24)(19,28,29,30)(23,31,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257381, #72257577, #72257648, …
maximal normal subgroup of	#72257381, #72257577, #72257648, …
maximal quotient of	#72307803, #72308610

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6441296);