#6401709

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	#2407	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1630 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)(17,18,23,24,22,28,19,26)(20,30,31,32,27,25,29,21);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(17,19,22,23)(18,24,28,26)(20,29,27,31)(21,30,32,25);(10,14)(13,16)(17,20)(18,25)(19,29)(21,24)(22,27)(23,31)(26,32)(28,30);(1,3,4,7)(2,6,5,8)(17,19,22,23)(18,26,28,24)(20,31,27,29)(21,30,32,25);(2,5)(6,8)(17,21,23,30,22,32,19,25)(18,27,24,29,28,20,26,31)

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,e^a=ef,d^b=dfi,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, 6308905);