#215735

order	512
nilpotency class	3
isoclinism	isoclinic class of #195972
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122931);