#509161

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#25805	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

795 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,h^d=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, 416357);