#251929

order	512
nilpotency class	3
isoclinism	isoclinic class of #251883
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	#29634	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1103 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)(19,22)(20,23)(21,24)(25,30)(26,29)(27,31)(28,32);(1,3,6,2)(4,7)(5,10,13,8)(9,12)(11,14)(15,16)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,30,31,28);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,31)(26,32)(29,30);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(20,27)(23,31)(25,32)(28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240625, #72240626, #72240639, …
maximal normal subgroup of	#72240625, #72240626, #72240639, …
maximal quotient of	#72251457, #72251467, #72251525

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159125);