#346455

order	512
nilpotency class	3
isoclinism	isoclinic class of #346446
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2335 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234505
maximal normal subgroup of	#72234505

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253651);