#346566

order	512
nilpotency class	3
isoclinism	isoclinic class of #346550
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62892	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,c^b=ch,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh,g^d=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268349, #72268355, #72268357, …
maximal normal subgroup of	#72268349, #72268355, #72268357, …
maximal quotient of	#72282064, #72282239

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253762);