#496266

order	512
nilpotency class	4
isoclinism	isoclinic class of #33758
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cei,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403462);