#235294

order	512
nilpotency class	3
isoclinism	isoclinic class of #235291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1245 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:

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

Polycyclic presentation:

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

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, 142490);