#251864

order	512
nilpotency class	3
isoclinism	isoclinic class of #251827
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1161 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)(3,6)(4,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,28)(27,30)(31,32);(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,19,21,26)(18,22,24,28)(20,25,27,31)(23,29,30,32);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(22,28)(29,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32)

Polycyclic presentation:

2^9|b^2=ei,d^2=hi,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	#72240624, #72240626, #72240627, …
maximal normal subgroup of	#72240624, #72240626, #72240627, …
maximal quotient of	#72251441, #72251502, #72251548, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159060);