#6493920

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37305	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1636 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18)(19,20)(21,32)(22,25)(23,29)(24,28)(26,27)(30,31);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,27,24,22,30,21,29)(18,23,32,31,25,28,26,20);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,21,28,22,31,27,23)(18,24,26,19,25,29,32,30);(10,13)(14,16)(17,18)(19,28)(20,24)(21,26)(22,25)(23,30)(27,32)(29,31);(9,11,12,15)(10,14,13,16)(18,25)(20,31)(23,28)(26,32)

Polycyclic presentation:

2^9|b^2=gi,c^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6401116);