#7732477

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

7099 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:

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

Polycyclic presentation:

2^9|a^2=gi,e^2=hi,f^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, 7639673);