#7813119

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

6763 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)(7,15)(8,12)(9,13)(10,14)(11,16)(17,18)(19,20)(21,22,26,31)(23,27,33,28)(24,32,35,29)(25,30,36,34);(1,2,3,4)(5,7,10,16)(6,11,14,15)(8,9,18,19)(12,20,17,13)(21,23)(22,27)(24,30)(25,29)(26,33)(28,31)(32,36)(34,35);(7,16)(9,19)(11,15)(13,20)(21,23,26,33)(22,28,31,27)(24,30,35,34)(25,29,36,32);(5,8)(6,12)(7,9)(10,18)(11,20)(13,15)(14,17)(16,19)(21,23)(22,27)(24,34)(25,32)(26,33)(28,31)(29,36)(30,35);(5,6)(7,11)(8,17)(9,13)(10,14)(12,18)(15,16)(19,20)(21,24,26,35)(22,29,31,32)(23,30,33,34)(25,28,36,27);(5,9)(6,13)(7,8)(10,19)(11,17)(12,15)(14,20)(16,18)(21,25)(22,30)(23,32)(24,28)(26,36)(27,35)(29,33)(31,34)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7720315);