#7710084

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6712 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)(8,15)(12,17)(13,19)(18,20)(21,22,28,32)(23,24,26,29)(25,30,36,34)(27,35,33,31);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(21,23,28,26)(22,29,32,24)(25,33,36,27)(30,35,34,31);(6,10)(11,16)(12,17)(18,20)(21,24,28,29)(22,23,32,26)(25,35,36,31)(27,34,33,30);(5,7,9,14)(6,11,10,16)(8,13,15,19)(12,18,17,20)(21,25,28,36)(22,30,32,34)(23,33,26,27)(24,31,29,35);(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(21,26,28,23)(22,24,32,29)(25,27,36,33)(30,31,34,35);(5,8)(6,12)(7,13)(9,15)(10,17)(11,18)(14,19)(16,20)(21,27)(22,31)(23,25)(24,34)(26,36)(28,33)(29,30)(32,35)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7617280);