#7729013

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

7005 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)(21,22,25,29)(23,26,31,27)(24,28,33,36)(30,34,32,35);(1,2)(3,4)(5,6)(7,12)(8,13)(9,17)(10,11)(14,18)(15,16)(19,20)(21,23)(22,26)(24,32)(25,31)(27,29)(28,35)(30,33)(34,36);(6,11)(8,12)(14,17)(16,19)(21,23,25,31)(22,27,29,26)(24,30,33,32)(28,35,36,34);(5,7,10,13)(6,12,11,8)(9,15,18,20)(14,19,17,16)(21,23)(22,26)(24,30)(25,31)(27,29)(28,34)(32,33)(35,36);(5,8,10,12)(6,13,11,7)(9,16,18,19)(14,20,17,15)(21,22)(23,26)(24,28)(25,29)(27,31)(30,34)(32,35)(33,36);(5,9)(6,14)(7,15)(8,16)(10,18)(11,17)(12,19)(13,20)(21,24)(22,28)(23,30)(25,33)(26,34)(27,35)(29,36)(31,32)

Polycyclic presentation:

2^9|a^2=h,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, 7636209);