#7724349

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:

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

Polycyclic presentation:

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

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, 7631545);