#509285

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24778	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416481);