#7657190

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	#37452	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6622 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)(5,6,10,8)(7,13,15,11)(9,12,17,16)(14,19,20,18)(21,22,25,28)(23,29,31,26)(24,27,32,34)(30,35,36,33);(1,2)(3,4)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(5,7,10,15)(6,11,8,13)(9,14,17,20)(12,18,16,19)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(5,8,10,6)(7,13,15,11)(9,12,17,16)(14,18,20,19)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)

Polycyclic presentation:

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

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, 7564386);