#483766

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#466	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

418 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321706, #72321994, #72322004
maximal normal subgroup of	#72321706, #72321994, #72322004

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390962);