#484127

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	C₂³	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

397 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=di,e^a=eg,g^a=gi,d^b=dh,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, 391323);