#190096

order	512
nilpotency class	3
isoclinism	isoclinic class of #190068
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

796 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97292);