#202508

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

757 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,f^c=fh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109704);