#7718011

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

6776 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)(5,6,9,11)(7,12,14,10)(8,16,17,13)(15,18,20,19)(21,22,26,24)(23,29,31,27)(25,28,33,32)(30,35,36,34);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,12)(13,18)(16,19)(17,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,34)(32,35)(33,36);(6,11)(7,14)(13,16)(15,20)(22,24)(27,29)(28,32)(34,35);(5,7)(6,12)(8,15)(9,14)(10,11)(13,19)(16,18)(17,20)(21,24,26,22)(23,29,31,27)(25,32,33,28)(30,35,36,34);(5,6)(7,12)(8,13)(9,11)(10,14)(15,19)(16,17)(18,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,34)(32,35)(33,36);(5,8,9,17)(6,13,11,16)(7,15,14,20)(10,18,12,19)(21,25,26,33)(22,28,24,32)(23,30,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7625207);