#7719105

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

(1,2,3,4)(5,6)(7,8)(9,13)(10,11)(12,16)(14,15)(17,18)(19,20)(23,26)(25,33)(28,30)(29,32);(1,2)(3,4)(21,22,27,24)(23,30,26,28)(25,32,33,29)(31,35,34,36);(6,11)(7,12)(13,18)(14,19)(21,23,27,26)(22,28,24,30)(25,34,33,31)(29,36,32,35);(5,7,10,12)(6,8,11,16)(9,14,17,19)(13,15,18,20)(21,24,27,22)(23,28,26,30)(25,32,33,29)(31,35,34,36);(5,8)(6,12)(7,11)(9,15)(10,16)(13,19)(14,18)(17,20)(21,25,27,33)(22,29,24,32)(23,31,26,34)(28,35,30,36);(5,9,10,17)(6,13,11,18)(7,14,12,19)(8,15,16,20)(21,26,27,23)(22,28,24,30)(25,34,33,31)(29,35,32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7626301);