#7658156

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

6928 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)(7,12)(8,11)(9,13)(10,15)(14,18)(16,19)(17,20)(21,22,28,31)(23,32,27,26)(24,29,36,33)(25,34,35,30);(1,2,3,4)(21,23,28,27)(22,26,31,32)(24,34,36,30)(25,33,35,29);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,24)(22,29)(23,30)(25,32)(26,35)(27,34)(28,36)(31,33);(5,8,9,16)(6,11,13,19)(7,14,15,20)(10,17,12,18)(21,25,28,35)(22,30,31,34)(23,33,27,29)(24,32,36,26);(5,7)(6,12)(8,14)(9,15)(10,13)(11,18)(16,20)(17,19)(21,26,28,32)(22,23,31,27)(24,35,36,25)(29,30,33,34);(5,6)(7,10)(8,11)(9,13)(12,15)(14,17)(16,19)(18,20)(21,27,28,23)(22,26,31,32)(24,34,36,30)(25,29,35,33)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7565352);