#222281

order	512
nilpotency class	3
isoclinism	isoclinic class of #222272
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

851 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 129477);