#442666

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=efgi,c^2=e,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,d^c=di

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, 349862);