#7710960

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

6790 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)(8,16)(12,17)(14,20)(18,19)(22,27)(24,32)(25,35)(28,31);(1,2)(3,4)(6,10)(11,13)(12,17)(18,19)(22,27)(28,31)(29,34)(30,36);(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,22,26,27)(23,31,33,28)(24,34,32,29)(25,30,35,36);(21,23,26,33)(22,28,27,31)(24,25,32,35)(29,30,34,36);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,24,26,32)(22,29,27,34)(23,25,33,35)(28,30,31,36);(5,8)(6,12)(7,14)(9,16)(10,17)(11,18)(13,19)(15,20)(21,25)(22,30)(23,32)(24,33)(26,35)(27,36)(28,34)(29,31)

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=i,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, 7618156);