#484034

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

426 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)(17,18,21,29,24,30,23,27)(19,31,33,41,35,47,40,46)(20,26,36,45,37,28,22,38)(25,39,43,32,44,48,42,34);(17,19)(18,25)(20,32)(21,33)(22,39)(23,40)(24,35)(26,41)(27,42)(28,46)(29,43)(30,44)(31,38)(34,37)(36,48)(45,47);(1,3,4,10,5,9,11,15)(2,6,7,13,8,12,14,16)(17,20,23,22,24,37,21,36)(18,26,27,38,30,28,29,45)(19,32,40,39,35,34,33,48)(25,41,42,31,44,46,43,47);(3,9)(6,12)(10,15)(13,16)(17,21,24,23)(18,27,30,29)(19,33,35,40)(20,36,37,22)(25,42,44,43)(26,38,28,45)(31,46,47,41)(32,48,34,39)

Polycyclic presentation:

2^9|a^2=f,c^2=fi,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	#72320258, #72321776, #72321833
maximal normal subgroup of	#72320258, #72321776, #72321833

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391230);