#325556

order	512
nilpotency class	3
isoclinism	isoclinic class of #279051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

554 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)(3,7)(4,6)(5,8)(9,10,12,15)(11,17,19,24)(13,21,18,16)(14,22,23,20)(25,26)(27,33)(28,31)(29,32)(30,36)(34,38)(35,40)(37,39);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,23)(22,24)(25,27,34,35,28,36,29,37)(26,30,38,39,31,33,32,40);(1,3,5,4)(2,7,8,6)(9,11,18,22,12,19,13,20)(10,14,21,24,15,23,16,17)(27,35)(29,34)(30,39)(32,38)(33,40)(36,37);(1,4,5,3)(2,7,8,6)(10,15)(14,23)(16,21)(17,24)(26,31)(27,36)(32,38)(35,37)

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=efi,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 232752);