#484051

order	512
nilpotency class	4
isoclinism	isoclinic class of #484039
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	#2184	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

432 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)(19,29,31,40)(20,33)(21,25)(22,37)(23,26)(24,30,41,47)(27,34)(28,39)(32,44,38,46)(35,45,48,42)(36,43);(17,19)(18,24)(20,30)(21,35)(22,38)(23,31)(25,40)(26,41)(27,44)(28,45)(29,43)(32,39)(33,46)(34,47)(36,48)(37,42);(1,3,4,10,5,9,11,15)(2,6,7,13,8,12,14,16)(17,20,22,21,23,34,39,36)(18,25,28,33,26,43,37,27)(19,30,38,35,31,47,32,48)(24,40,45,46,41,29,42,44);(3,9)(6,12)(10,15)(13,16)(18,26)(24,41)(25,43)(27,33)(28,37)(29,40)(42,45)(44,46)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216779, #72216810, #72216824, …
maximal normal subgroup of	#72216779, #72216810, #72216824, …
maximal quotient of	C₂×#484051, #72318551, #72318592, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391247);