#7912281

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

6829 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)(7,8)(9,17)(10,14)(11,12)(13,19)(15,20)(16,18)(22,26)(23,29)(28,34)(30,31);(1,2,3,4)(5,7,11,10)(6,8,12,14)(9,15,19,18)(13,16,17,20)(21,22)(23,27)(24,28)(25,26)(29,33)(30,35)(31,36)(32,34);(6,12)(8,14)(13,17)(16,20)(21,22,25,26)(23,27,29,33)(24,28,32,34)(30,35,31,36);(5,8,11,14)(6,7,12,10)(9,16,19,20)(13,15,17,18)(21,22)(23,27)(24,28)(25,26)(29,33)(30,35)(31,36)(32,34);(5,9,11,19)(6,13,12,17)(7,15,10,18)(8,16,14,20)(21,23,25,29)(22,27,26,33)(24,31,32,30)(28,36,34,35);(5,10,11,7)(6,14,12,8)(9,18,19,15)(13,20,17,16)(21,24,25,32)(22,28,26,34)(23,30,29,31)(27,35,33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7819477);