#412144

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3153	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 319340);