#7645729

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	C₂³×D₄	C₂³
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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=h,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, 7552925);