#484036

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	#2186	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

425 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260325, #72260357, #72260372
maximal normal subgroup of	#72260325, #72260357, #72260372

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391232);