#7500853

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	C₂×#3010	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3196 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)(9,10,15,20,16,21,14,19)(11,22,26,36,27,35,25,17)(12,28,29,33,18,13,30,32)(23,24,34,31,37,38,39,40);(5,6)(7,8)(9,11,16,27)(10,17,21,36)(12,23,18,37)(13,31,28,40)(14,26,15,25)(19,22,20,35)(24,33,38,32)(29,39,30,34);(2,4)(6,8)(9,12,16,18)(10,13,21,28)(11,23,27,37)(14,30,15,29)(17,31,36,40)(19,33,20,32)(22,38,35,24)(25,39,26,34);(2,4)(9,13)(10,18)(11,24)(12,21)(14,32)(15,33)(16,28)(17,34)(19,30)(20,29)(22,23)(25,31)(26,40)(27,38)(35,37)(36,39);(9,14,16,15)(10,19,21,20)(11,25,27,26)(12,29,18,30)(13,32,28,33)(17,35,36,22)(23,34,37,39)(24,31,38,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^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, 7408049);