#7656012

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

7080 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,e^2=h,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, 7563208);