#7505361

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	#23426	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1910 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)(19,26)(20,22)(21,31)(23,25)(24,28)(27,29)(30,32);(1,3,6,8)(2,4,7,5)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,26)(21,30)(22,28)(23,29)(25,32)(27,31);(1,4,6,5)(2,3,7,8)(10,14)(13,16)(17,20,22,18)(19,27,28,32)(21,31,23,25)(24,29,26,30);(1,5,6,4)(2,3,7,8)(10,14)(13,16)(19,28)(24,26)(27,32)(29,30);(17,21,22,23)(18,25,20,31)(19,29,28,30)(24,27,26,32)

Polycyclic presentation:

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

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, 7412557);