#6460052

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444408
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	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2152 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=f,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^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, 6367248);