#498692

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

513 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ef,d^2=ef,g^2=h,h^2=i,b^a=be,c^a=cf,d^b=dei,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405888);