#7507773

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1865 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 7414969);