#245936

order	512
nilpotency class	3
isoclinism	isoclinic class of #245903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1334 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,e^b=eh,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219481
maximal normal subgroup of	#72219481

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153132);