#6402749

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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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)(4,6)(7,8)(9,10,12,13)(11,14,15,16)(17,18,24,29,23,28,21,26)(19,30,20,31,27,25,32,22);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(17,19,23,27)(18,25,28,30)(20,21,32,24)(22,26,31,29);(2,6)(5,8)(17,20)(18,22)(19,21)(23,32)(24,27)(25,26)(28,31)(29,30);(9,11,12,15)(10,14,13,16)(17,21,23,24)(18,26,28,29)(19,20,27,32)(22,30,31,25);(10,13)(14,16)(17,22,21,25,23,31,24,30)(18,27,26,20,28,19,29,32)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,e^a=ef,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, 6309945);