#498770

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 56 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,29,33,30,34,41,47,28,32,39,45,40,46,52,55)(27,35,37,48,38,49,51,56,36,31,42,43,50,44,53,54);(3,8)(5,7)(9,11,14,21)(10,15,17,18)(12,20,13,19)(16,24,22,23);(1,3,4,8)(2,5,6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,28,36)(26,31,32,35)(29,37,39,42)(30,38,40,50)(33,43,45,48)(34,44,46,49)(41,51,52,53)(47,54,55,56);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(27,36)(31,35)(37,42)(38,50)(43,48)(44,49)(51,53)(54,56)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=ef,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cefi,d^b=de,d^c=di

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, 405966);