#7660654

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	#52169	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6400 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,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, 7567850);