#7505365

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1937 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:

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

Polycyclic presentation:

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

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, 7412561);