#7648460

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×#3480	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6610 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7555656);