#7981531

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

6701 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)(5,6,10,13)(7,8,15,11)(9,12,19,20)(14,16,17,18)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(1,2,3,4)(5,7)(6,11)(8,13)(9,17)(10,15)(12,16)(14,19)(18,20)(22,26)(27,29)(28,32)(34,35);(5,7,10,15)(6,11,13,8)(9,14,19,17)(12,18,20,16)(21,22)(23,27)(24,32)(25,26)(28,33)(29,31)(30,35)(34,36);(5,7)(6,11)(8,13)(9,14)(10,15)(12,18)(16,20)(17,19)(21,23,25,31)(22,27,26,29)(24,30,33,36)(28,34,32,35);(5,8,10,11)(6,7,13,15)(9,18,19,16)(12,17,20,14)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36);(5,9)(6,12)(7,14)(8,16)(10,19)(11,18)(13,20)(15,17)(21,22)(23,27)(24,32)(25,26)(28,33)(29,31)(30,35)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7888727);