#251898

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

1099 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)(17,18,21,24)(19,22,26,29)(20,23,27,31)(25,30,32,28);(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,19)(18,22)(20,25)(21,26)(23,30)(24,29)(27,32)(28,31);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,31)(26,32)(29,30);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(20,27)(23,31)(25,32)(28,30)

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=h,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	#72283625, #72283707, #72283847, …
maximal normal subgroup of	#72283625, #72283707, #72283847, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159094);