#379377

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
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

750 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,29,24,30,22,28)(19,25,21,37,33,40,32,27)(20,26,35,43,36,44,38,45)(31,39,34,41,46,48,47,42);(1,3,5,4)(2,6,8,7)(10,13)(14,15)(17,19,22,32,24,33,23,21)(18,25,28,27,30,40,29,37)(20,34,35,46,36,47,38,31)(26,41,43,48,44,42,45,39);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,14,13,15)(17,20)(18,26)(19,31)(21,34)(22,38)(23,35)(24,36)(25,39)(27,42)(28,45)(29,43)(30,44)(32,47)(33,46)(37,41)(40,48);(1,4,5,3)(2,7,8,6)(10,13)(14,15)(17,21,23,33,24,32,22,19)(18,27,29,25,30,37,28,40)(20,34,35,46,36,47,38,31)(26,42,43,39,44,41,45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=e,d^2=efh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,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, 286573);