#7899485

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

6714 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7806681);