#7500849

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi

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, 7408045);