#232192

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,6,10,5,9,4,8)(3,11,12,16,13,7,14,15)(17,18)(19,23,29,39)(20,31,34,41)(21,27)(22,26)(24,35,42,46)(25,33)(28,40)(30,38,36,43)(32,37)(44,48)(45,47);(17,19,21,29)(18,23,27,39)(20,32,34,45)(22,36,33,30)(24,40,42,48)(25,38,26,43)(28,35,44,46)(31,37,41,47);(1,3,5,13)(2,7,9,11)(4,14,6,12)(8,16,10,15)(17,20,33,46,21,34,22,35)(18,24,26,31,27,42,25,41)(19,28,36,32,29,44,30,45)(23,37,38,40,39,47,43,48);(1,4,5,6)(2,8,9,10)(3,12,13,14)(7,15,11,16)(18,25,27,26)(19,29)(23,38,39,43)(24,41,42,31)(28,44)(30,36)(32,45)(37,40,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,e^a=eh,c^b=cefi,d^b=de,f^b=fh,g^b=gh,d^c=di,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139388);