#7513495

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,d^2=fi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,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, 7420691);