#7652768

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

6684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=i,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7559964);