#498700

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

510 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)(17,18,22,27,23,19,28,31,21,26,37,44,38,30,24,39)(20,32,35,46,36,33,42,47,34,25,41,43,48,29,40,45);(1,2)(3,7)(4,8)(5,6)(9,11)(10,13)(12,16)(14,15)(17,19,22,31,23,26,28,44,21,30,37,39,38,18,24,27)(20,33,35,47,36,25,42,43,34,29,41,45,48,32,40,46);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,36)(24,40)(26,32)(27,43)(28,42)(30,33)(31,45)(37,41)(38,48)(39,47)(44,46);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(20,34)(25,32)(29,33)(35,41)(36,48)(40,42)(43,46)(45,47)

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,c^b=ci,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, 405896);