#6457716

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444406
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=fgi,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei

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, 6364912);