#6431451

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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	#2676	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1398 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)(3,8)(4,7)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,21,25,22,26,30,32)(19,23,20,28,27,31,29,24);(2,6)(3,4)(9,10)(11,14)(12,13)(15,16)(19,27)(20,29)(23,31)(24,28);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,19,21,20,22,27,30,29)(18,23,25,28,26,31,32,24);(1,4,5,3)(2,8,6,7)(9,11,12,16)(10,14,13,15)(19,27)(20,29)(23,31)(24,28);(2,6)(7,8)(17,20,21,27,22,29,30,19)(18,24,25,23,26,28,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6338647);