#7501835

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500740
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	#60644	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2340 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)(3,5)(4,6)(7,8)(9,10,14,20)(11,21,24,16)(12,26,29,36)(13,31,15,19)(17,28,37,30)(18,25,33,23)(22,34,39,35)(27,38,40,32);(1,3,5,2)(4,7,8,6)(9,11,14,24)(10,16,20,21)(12,27,29,40)(13,25,15,23)(17,34,37,35)(18,19,33,31)(22,30,39,28)(26,32,36,38);(1,4,5,8)(2,6,3,7)(9,12,14,29)(10,17,20,37)(11,22,24,39)(13,28,15,30)(16,32,21,38)(18,35,33,34)(19,36,31,26)(23,27,25,40);(2,3)(6,7)(9,11)(10,18)(12,22)(13,23)(14,24)(15,25)(16,31)(17,35)(19,21)(20,33)(26,32)(27,28)(29,39)(30,40)(34,37)(36,38);(9,13,14,15)(10,19,20,31)(11,23,24,25)(12,28,29,30)(16,18,21,33)(17,36,37,26)(22,27,39,40)(32,35,38,34)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409031);