#7500838

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
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	C₂×#3008	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3318 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)(9,10,15,30,14,20,13,19)(11,21,29,16,25,36,24,32)(12,17,26,18,28,34,27,33)(22,35,23,31,39,37,38,40);(5,6)(7,8)(9,11)(10,16)(12,22)(13,29)(14,25)(15,24)(17,31)(18,35)(19,36)(20,32)(21,30)(23,27)(26,38)(28,39)(33,37)(34,40);(2,4)(6,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,26)(16,31)(18,30)(19,33)(20,34)(21,35)(23,29)(24,38)(25,39)(32,40)(36,37);(2,4)(9,12)(10,18)(11,23)(13,26)(14,28)(15,27)(16,31)(17,30)(19,34)(20,33)(21,37)(22,29)(24,39)(25,38)(32,40)(35,36);(9,13,14,15)(10,19,20,30)(11,24,25,29)(12,26,28,27)(16,21,32,36)(17,18,34,33)(22,23,39,38)(31,37,40,35)

Polycyclic presentation:

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

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, 7408034);