#7719106

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	#35210	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7626302);