#406890

order	512
nilpotency class	3
isoclinism	isoclinic class of #395154
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

918 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=fi,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^c=di

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, 314086);