#191270

order	512
nilpotency class	3
isoclinism	isoclinic class of #191240
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

502 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,g^b=ghi,g^c=ghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98466);