#7651277

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×#3494	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6773 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)(5,6)(7,13)(8,12)(9,10)(11,15)(14,20)(16,19)(17,18)(21,22)(23,33)(24,29)(25,30)(26,31)(27,32)(28,34)(35,36);(1,2,3,4)(21,23)(22,28)(24,26)(25,31)(27,34)(29,36)(30,35)(32,33);(6,10)(11,13)(12,18)(19,20)(23,34)(26,35)(28,33)(31,36);(5,7,9,15)(6,11,10,13)(8,16,17,14)(12,20,18,19)(21,24)(22,29)(23,26)(25,32)(27,30)(28,36)(31,33)(34,35);(5,8,9,17)(6,12,10,18)(7,14,15,16)(11,19,13,20)(21,25,27,29)(22,30,32,24)(23,31,34,36)(26,28,35,33);(5,6)(7,11)(8,12)(9,10)(13,15)(14,19)(16,20)(17,18)(21,26)(22,31)(23,24)(25,28)(27,35)(29,33)(30,34)(32,36)

Polycyclic presentation:

2^9|b^2=gi,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7558473);