#481319

order	512
nilpotency class	4
isoclinism	isoclinic class of #481319
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#60695	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

754 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)(17,18,21,26,22,28,36,27)(19,24,31,42,29,40,45,43)(20,32,34,25,35,23,33,38)(30,39,46,41,44,37,47,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(18,23)(19,29)(20,33)(21,36)(24,39)(25,26)(27,38)(28,32)(30,46)(34,35)(37,40)(41,43)(42,48)(44,47);(3,9)(6,12)(10,15)(13,16)(17,19)(18,24)(20,30)(21,31)(22,29)(23,37)(25,41)(26,42)(27,43)(28,40)(32,39)(33,47)(34,46)(35,44)(36,45)(38,48);(19,29)(24,40)(30,44)(31,45)(37,39)(41,48)(42,43)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216726, #72216740, #72216898, …
maximal normal subgroup of	#72216726, #72216740, #72216898, …
maximal quotient of	C₂×#481319, #72318675, #72318781, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388515);