#7729014

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

6816 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)(7,14)(8,12)(9,11)(10,13)(15,20)(16,18)(17,19)(21,22,25,29)(23,26,31,27)(24,28,33,36)(30,34,32,35);(1,2,3,4)(5,7,10,9)(6,11,13,14)(8,15,17,16)(12,18,19,20)(21,23)(22,26)(24,32)(25,31)(27,29)(28,35)(30,33)(34,36);(7,9)(11,14)(15,16)(18,20)(21,23,25,31)(22,27,29,26)(24,30,33,32)(28,35,36,34);(5,8,10,17)(6,12,13,19)(7,15,9,16)(11,18,14,20)(21,23)(22,26)(24,30)(25,31)(27,29)(28,34)(32,33)(35,36);(5,9,10,7)(6,11,13,14)(8,16,17,15)(12,18,19,20)(21,22)(23,26)(24,28)(25,29)(27,31)(30,34)(32,35)(33,36);(5,6)(7,14)(8,12)(9,11)(10,13)(15,20)(16,18)(17,19)(21,24)(22,28)(23,30)(25,33)(26,34)(27,35)(29,36)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=hi,e^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7636210);