#7502383

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	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2094 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)(3,7,8,6)(9,10,16,20)(11,17,23,12)(13,28,30,36)(14,31,15,21)(18,27,24,26)(19,32,35,29)(22,37,38,40)(25,34,33,39);(2,5)(3,8)(9,11)(10,17)(12,20)(13,22)(14,24)(15,18)(16,23)(19,33)(21,27)(25,35)(26,31)(28,37)(29,34)(30,38)(32,39)(36,40);(9,12,16,17)(10,18,20,24)(11,21,23,31)(13,25,30,33)(14,27,15,26)(19,34,35,39)(22,36,38,28)(29,37,32,40);(1,3,4,8)(2,6,5,7)(9,13)(10,19)(11,22)(12,25)(14,32)(15,29)(16,30)(17,33)(18,34)(20,35)(21,36)(23,38)(24,39)(26,37)(27,40)(28,31);(10,20)(11,23)(18,24)(19,35)(21,31)(22,38)(28,36)(34,39)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,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, 7409579);