#484018

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

419 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391214);