#241752

order	512
nilpotency class	3
isoclinism	isoclinic class of #232215
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1310 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)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,26)(20,24)(21,23)(22,27)(25,29)(28,31)(30,32);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,22)(20,27)(21,28)(23,30)(24,26)(25,31)(29,32);(13,16)(18,23)(21,29)(22,30)(24,25)(26,31)(28,32);(1,3,4,7)(2,6,5,8)(19,27)(22,26)(28,32)(30,31)

Polycyclic presentation:

2^9|d^2=eh,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309269, #72309272, #72309278, …
maximal normal subgroup of	#72309269, #72309272, #72309278, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148948);