#8012018

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

6850 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)(7,17)(8,13)(9,14)(10,15)(11,16)(12,18)(19,20)(21,22,27,24)(23,30,25,28)(26,32,34,29)(31,35,33,36);(1,2,3,4)(5,7)(6,12)(8,10)(9,15)(11,18)(13,20)(14,19)(16,17)(21,22,27,24)(23,28,25,30)(26,29,34,32)(31,35,33,36);(7,18)(10,19)(12,17)(15,20)(21,23,27,25)(22,28,24,30)(26,31,34,33)(29,35,32,36);(5,8)(6,13)(7,10)(9,16)(11,14)(12,20)(15,17)(18,19)(21,24,27,22)(23,28,25,30)(26,32,34,29)(31,35,33,36);(5,9,11,13)(6,14,16,8)(7,15,18,20)(10,12,19,17)(21,25,27,23)(22,28,24,30)(26,33,34,31)(29,35,32,36);(5,10)(6,15)(7,8)(9,12)(11,19)(13,17)(14,18)(16,20)(21,26,27,34)(22,29,24,32)(23,31,25,33)(28,35,30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7919214);