#379436

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=ef,d^2=fh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286632);