#323496

order	512
nilpotency class	3
isoclinism	isoclinic class of #298795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=eghi,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230692);