#7655068

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

6630 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)(7,12)(8,16)(9,13)(10,15)(11,17)(14,18)(19,20)(21,22,26,29)(23,27,30,34)(24,25,28,33)(31,32,35,36);(1,2,3,4)(23,30)(27,34)(31,35)(32,36);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,26,30)(22,27,29,34)(24,31,28,35)(25,32,33,36);(5,8,9,17)(6,11,13,16)(7,14,15,20)(10,18,12,19);(5,7)(6,12)(8,14)(9,15)(10,13)(11,19)(16,18)(17,20)(21,24)(22,25)(23,31)(26,28)(27,32)(29,33)(30,35)(34,36);(5,6)(7,10)(8,11)(9,13)(12,15)(14,18)(16,17)(19,20)(21,25)(22,28)(23,32)(24,29)(26,33)(27,35)(30,36)(31,34)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=h,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, 7562264);