#483885

order	512
nilpotency class	4
isoclinism	isoclinic class of #483882
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33675	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

438 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dh,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216788, #72216809, #72216810, …
maximal normal subgroup of	#72216788, #72216809, #72216810, …
maximal quotient of	C₂×#483885, #72318551, #72318613, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391081);