#484129

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	#2215	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

414 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)(17,18,23,27,24,30,21,29)(19,31,33,46,35,47,34,41)(20,28,36,45,37,26,22,38)(25,40,43,48,44,39,42,32);(1,3)(2,7)(4,9)(5,11)(6,12)(8,10)(13,16)(14,15)(17,19)(18,25)(20,32)(21,34)(22,39)(23,33)(24,35)(26,41)(27,43)(28,46)(29,42)(30,44)(31,38)(36,40)(37,48)(45,47);(1,4,6,10,5,13,14,7)(2,3,9,12,8,11,16,15)(17,20,23,36,24,37,21,22)(18,26,27,38,30,28,29,45)(19,32,33,40,35,48,34,39)(25,41,43,31,44,46,42,47);(3,11)(4,13)(7,10)(12,15)(17,21,24,23)(18,27,30,29)(19,33,35,34)(20,22,37,36)(25,42,44,43)(26,38,28,45)(31,41,47,46)(32,40,48,39)

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,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, 391325);