#346128

order	512
nilpotency class	3
isoclinism	isoclinic class of #346096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

467 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)(3,6)(4,8)(5,7)(9,13)(10,15)(11,14)(12,16)(17,18,23,28)(19,29,32,24)(20,34,37,46)(21,27,22,26)(25,39,44,36)(30,43,48,41)(31,47,35,40)(33,42,38,45);(1,3)(2,6)(4,9)(5,11)(7,14)(8,13)(10,12)(15,16)(17,19,21,38,23,32,22,33)(18,24,26,45,28,29,27,42)(20,35,36,48,37,31,39,30)(25,43,34,47,44,41,46,40);(3,9)(4,12)(6,13)(8,16)(10,11)(14,15)(17,20)(18,25)(19,30,32,48)(21,39)(22,36)(23,37)(24,40,29,47)(26,46)(27,34)(28,44)(31,33,35,38)(41,42,43,45);(2,7)(6,14)(8,16)(13,15)(17,20)(18,25)(19,31)(21,39)(22,36)(23,37)(24,41)(26,46)(27,34)(28,44)(29,43)(30,33)(32,35)(38,48)(40,42)(45,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=defi,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253324);