#376899

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1547 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)(5,6,8,7)(9,10,15,19)(11,16,22,33)(12,23,25,36)(13,27,14,18)(17,29,35,26)(20,34,37,40)(21,32,28,31)(24,39,38,30);(2,4)(9,11)(10,16)(12,24)(13,21)(14,28)(15,22)(17,34)(18,31)(19,33)(20,26)(23,39)(25,38)(27,32)(29,37)(30,36)(35,40);(2,4)(9,12,13,26,15,25,14,29)(10,17,18,23,19,35,27,36)(11,20,21,38,22,37,28,24)(16,30,31,34,33,39,32,40);(2,4)(5,7,8,6)(12,25)(17,35)(20,37)(23,36)(24,38)(26,29)(30,39)(34,40)

Polycyclic presentation:

2^9|a^2=h,c^2=ef,d^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,d^c=di,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, 284095);