#484016

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 ◁ C₄×M₁₆ ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

424 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)(19,28)(20,26)(21,33)(22,37)(23,25)(24,30)(27,35)(29,46)(31,45)(32,42)(34,40)(36,44)(38,43)(39,47)(41,48);(17,19,23,31)(18,24,25,39)(20,29,22,38)(21,34,36,48)(26,40,37,41)(27,43,42,46)(28,44,45,33)(30,32,47,35);(1,3,4,10,5,9,11,15)(2,6,7,13,8,12,14,16)(18,25)(24,39)(26,37)(27,42)(28,45)(33,44)(40,41)(43,46);(3,9)(6,12)(10,15)(13,16)(17,20,23,22)(18,26,25,37)(19,29,31,38)(21,32,36,35)(24,40,39,41)(27,33,42,44)(28,46,45,43)(30,34,47,48)

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,c^a=ci,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234336, #72260357, #72260370
maximal normal subgroup of	#72234336, #72260357, #72260370

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391212);