#7500856

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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×#3012	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3266 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)(9,10,15,18,14,19,13,20)(11,21,23,16,24,34,25,30)(12,17,28,31,26,33,27,32)(22,35,36,29,38,40,37,39);(5,6)(7,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,29)(18,21)(19,30)(20,34)(26,38)(27,36)(28,37)(31,35)(32,40)(33,39);(2,4)(6,8)(9,12)(10,17)(11,22)(13,27)(14,26)(15,28)(16,29)(18,31)(19,33)(20,32)(21,35)(23,36)(24,38)(25,37)(30,39)(34,40);(2,4)(10,18)(11,23)(13,15)(17,31)(19,20)(21,34)(22,36)(24,25)(27,28)(32,33)(35,40)(37,38);(10,19)(11,24)(12,26)(18,20)(23,25)(27,28)(29,39)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408052);