#7647184

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

6700 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)(6,10)(8,16)(11,13)(14,20)(21,22)(23,32)(24,30)(25,27)(26,29)(28,31)(33,36)(34,35);(1,2,3,4)(21,23,28,26)(22,29,31,32)(24,35,25,33)(27,34,30,36);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,24,28,25)(22,30,31,27)(23,33,26,35)(29,34,32,36);(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,25,28,24)(22,27,31,30)(23,33,26,35)(29,34,32,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,26,28,23)(22,29,31,32)(24,35,25,33)(27,36,30,34);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,27)(22,24)(23,34)(25,31)(26,36)(28,30)(29,35)(32,33)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7554380);