#225506

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

558 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132702);