#7658097

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×#3515	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6388 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)(6,10)(11,17)(12,18)(19,20)(21,22,26,25)(23,29,30,27)(24,32,33,28)(31,34,36,35);(1,2,3,4)(23,30)(27,29)(31,36)(34,35);(5,6,9,10)(7,11,14,17)(8,12,16,18)(13,19,15,20)(21,23,26,30)(22,27,25,29)(24,31,33,36)(28,34,32,35);(21,24,26,33)(22,28,25,32)(23,31,30,36)(27,34,29,35);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,22,26,25)(23,27,30,29)(24,28,33,32)(31,34,36,35);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,26,22)(23,29,30,27)(24,32,33,28)(31,35,36,34)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=i,e^2=h,f^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, 7565293);