#6788304

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1365 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)(17,18,23,26)(19,28,30,24)(20,32,27,22)(21,29,31,25);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,24,26,28)(20,29,27,25)(21,22,31,32);(2,5)(6,8)(10,13)(14,16)(17,20,22,18,23,27,32,26)(19,29,31,24,30,25,21,28);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,22)(20,24)(23,31)(26,29)(27,28)(30,32);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,22,23,32)(18,20,26,27)(19,21,30,31)(24,25,28,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6695500);