#474692

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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 ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1002 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)(5,6,10,9)(7,12,8,11)(21,22,27,31)(23,32,24,35)(25,29,26,30)(28,34,36,33);(5,7,10,8)(6,11,9,12)(13,14,16,18)(15,19,20,17)(21,23,26,34,27,24,25,33)(22,28,29,35,31,36,30,32);(2,4)(5,8,10,7)(6,11,9,12)(13,15,16,20)(14,17,18,19)(22,29)(23,33)(24,34)(25,26)(30,31)(32,35);(5,9,10,6)(7,12,8,11)(21,24,26,33,27,23,25,34)(22,28,29,35,31,36,30,32)

Polycyclic presentation:

2^9|a^2=ef,b^2=egh,c^2=eg,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,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, 381888);