#6401685

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2671	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1562 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,27,23,22,21,32)(19,28,20,30,29,25,31,26);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(17,19,23,29)(18,25,22,28)(20,21,31,24)(26,32,30,27);(10,14)(13,16)(17,20)(18,26)(19,21)(22,30)(23,31)(24,29)(25,32)(27,28);(1,3,4,7)(2,6,5,8)(17,21,23,24)(18,27,22,32)(19,20,29,31)(25,30,28,26);(2,5)(6,8)(17,22,24,32,23,18,21,27)(19,25,20,26,29,28,31,30)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,d^b=df,e^b=ei,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6308881);