#7711802

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

6644 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)(7,13)(8,16)(9,11)(10,14)(12,17)(15,18)(19,20)(25,33)(30,34)(31,36)(32,35);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,13)(12,18)(16,19)(17,20)(21,22)(23,28)(24,29)(25,30)(26,27)(31,35)(32,36)(33,34);(6,11)(7,14)(12,16)(15,20)(22,27)(24,28)(30,34)(32,35);(5,6)(7,10)(8,12)(9,11)(13,14)(15,18)(16,17)(19,20)(21,23,26,29)(22,28,27,24)(25,31,33,36)(30,35,34,32);(5,7)(6,10)(8,15)(9,14)(11,13)(12,18)(16,19)(17,20)(21,24,26,28)(22,29,27,23)(25,32,33,35)(30,36,34,31);(5,8,9,17)(6,12,11,16)(7,15,14,20)(10,18,13,19)(21,25)(22,30)(23,31)(24,32)(26,33)(27,34)(28,35)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618998);