#7658970

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×#3517	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6640 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)(6,12)(8,18)(13,15)(16,19)(21,22,27,26)(23,31,34,28)(24,29,33,35)(25,30,32,36);(1,2)(3,4)(5,6)(7,15)(8,10)(9,13)(11,12)(14,18)(16,20)(17,19)(21,22,27,26)(23,28,34,31)(24,35,33,29)(25,30,32,36);(5,7,11,9)(6,13,12,15)(8,16,18,19)(10,17,14,20)(21,23)(22,28)(24,32)(25,33)(26,31)(27,34)(29,30)(35,36);(5,8)(6,14)(7,16)(9,19)(10,12)(11,18)(13,20)(15,17)(21,24,27,33)(22,29,26,35)(23,32,34,25)(28,30,31,36);(5,9,11,7)(6,13,12,15)(8,19,18,16)(10,17,14,20)(21,25,27,32)(22,30,26,36)(23,33,34,24)(28,29,31,35);(5,10,11,14)(6,8,12,18)(7,17,9,20)(13,16,15,19)(21,26,27,22)(23,31,34,28)(24,35,33,29)(25,30,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566166);