#7766329

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
prime radical	C₂×#3535	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6581 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)(23,32)(25,35)(28,33)(29,34);(1,2)(3,4)(5,6,10,11)(7,12,17,20)(8,13,18,16)(9,14,19,15)(21,22,27,31)(23,33,32,28)(24,26,30,36)(25,29,35,34);(6,11)(9,19)(12,20)(13,16)(21,23,27,32)(22,28,31,33)(24,35,30,25)(26,29,36,34);(5,7)(6,12)(8,14)(9,16)(10,17)(11,20)(13,19)(15,18)(21,24,27,30)(22,26,31,36)(23,25,32,35)(28,34,33,29);(5,8)(6,13)(7,15)(9,12)(10,18)(11,16)(14,17)(19,20)(21,25)(22,29)(23,24)(26,33)(27,35)(28,36)(30,32)(31,34);(5,9)(6,14)(7,16)(8,12)(10,19)(11,15)(13,17)(18,20)(21,26)(22,30)(23,34)(24,31)(25,33)(27,36)(28,35)(29,32)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=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, 7673525);