#332402

order	512
nilpotency class	3
isoclinism	isoclinic class of #332327
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

572 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)(3,11,4,13)(5,15,16,9)(7,14,8,12)(17,18)(19,27)(20,31)(21,35)(22,26)(23,29)(24,33)(25,38)(28,46)(30,43)(32,41)(34,44)(36,39)(37,42)(40,47)(45,48);(17,19)(18,23)(20,32)(21,36)(22,30)(24,41)(25,43)(26,39)(27,35)(28,33)(29,38)(31,40)(34,48)(37,47)(42,45)(44,46);(1,3,6,4)(2,7,10,8)(5,14,16,12)(9,13,15,11)(23,39)(27,43)(40,46)(41,45);(1,4,6,3)(2,8,10,7)(5,12,16,14)(9,11,15,13)(17,20,22,34)(18,24,26,42)(19,28,30,47)(21,37,38,33)(23,40,39,46)(25,44,35,31)(27,45,43,41)(29,48,36,32)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239598);