#6412368

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#52164	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1754 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,21,25,22,19,23,27)(20,24,29,31,28,26,30,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14);(2,6)(3,7)(9,11,13,12)(10,15,14,16)(17,19,21,27,22,18,23,25)(20,26,29,32,28,24,30,31);(2,6)(5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(9,12,13,11)(10,15,14,16)(20,28)(24,26)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=h,c^2=fh,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,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, 6319564);