#510853

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

521 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ef,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418049);