#7720022

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×#3475	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6620 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)(5,6)(7,12)(8,10)(9,16)(11,14)(13,17)(15,18)(19,20)(24,32)(28,33)(31,35)(34,36);(1,2)(3,4)(5,7)(6,11)(8,12)(9,15)(10,14)(13,18)(16,19)(17,20)(22,26)(23,29)(28,33)(31,35);(6,8)(7,14)(13,16)(15,20)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(5,7)(6,12)(8,11)(9,15)(10,14)(13,19)(16,18)(17,20);(5,8,10,6)(7,11,14,12)(9,16,17,13)(15,18,20,19)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(5,9,10,17)(6,13,8,16)(7,15,14,20)(11,18,12,19)(21,24,25,32)(22,28,26,33)(23,31,29,35)(27,34,30,36)

Polycyclic presentation:

2^9|a^2=gh,e^2=h,f^2=i,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, 7627218);