#481412

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×#102
prime radical	#61433	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

715 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)(17,18,21,27,23,25,22,37)(19,28,30,46,31,45,38,43)(20,32,34,42,36,44,35,26)(24,33,40,47,39,48,41,29);(17,19,23,31)(18,24,25,39)(20,33,36,48)(21,30,22,38)(26,43,42,46)(27,40,37,41)(28,44,45,32)(29,34,47,35);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(18,25)(24,39)(26,42)(27,37)(28,45)(32,44)(40,41)(43,46);(3,9)(6,12)(10,15)(13,16)

Polycyclic presentation:

2^9|a^2=f,b^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=dhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388608);