#251858

order	512
nilpotency class	3
isoclinism	isoclinic class of #251827
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#10336	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1150 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283625, #72283654, #72283729, …
maximal normal subgroup of	#72283625, #72283654, #72283729, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159054);