#407930

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1027 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=ei,d^2=fi,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,h^a=hi,c^b=cg,g^b=gi,g^c=gi

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, 315126);