#7653881

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6588 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)(5,6,11,9)(7,13,8,12)(10,14,18,17)(15,20,16,19);(1,2,3,4)(5,6,11,9)(7,12,8,13)(10,14,18,17)(15,19,16,20)(22,26)(27,33)(28,34)(35,36);(5,7,11,8)(6,12,9,13)(10,15,18,16)(14,19,17,20)(21,22)(23,27)(24,28)(25,26)(29,35)(30,33)(31,36)(32,34);(5,8,11,7)(6,13,9,12)(10,16,18,15)(14,20,17,19);(5,9,11,6)(7,13,8,12)(10,14,18,17)(15,19,16,20)(21,23,25,30)(22,27,26,33)(24,31,32,29)(28,36,34,35);(5,10,11,18)(6,14,9,17)(7,15,8,16)(12,19,13,20)(21,24,25,32)(22,28,26,34)(23,29,30,31)(27,35,33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561077);