#358639

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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 ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

832 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,28,23,26,21,27)(19,24,30,39,31,38,35,43)(20,25,33,42,34,41,36,44)(29,37,45,48,46,47,32,40);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,32)(21,35)(22,30)(23,31)(25,40)(26,38)(27,43)(28,39)(29,33)(34,45)(36,46)(37,42)(41,48)(44,47);(9,12,13,11)(10,14,15,16)(17,20)(18,25)(19,29)(21,36)(22,33)(23,34)(24,37)(26,41)(27,44)(28,42)(30,45)(31,46)(32,35)(38,47)(39,48)(40,43);(1,3,4,7)(2,6,5,8)(18,26)(24,38)(25,41)(27,28)(37,47)(39,43)(40,48)(42,44)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265835);