#7655273

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6688 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)(6,11)(7,13)(14,19)(15,20);(1,2)(3,4)(21,22,26,29)(23,30,33,27)(24,25,28,34)(31,36,35,32);(5,6)(7,12)(8,13)(9,14)(10,11)(15,17)(16,20)(18,19)(21,23,26,33)(22,27,29,30)(24,31,28,35)(25,32,34,36);(5,7,10,13)(6,12,11,8)(9,15,18,20)(14,17,19,16);(5,8,10,12)(6,13,11,7)(9,17,18,16)(14,15,19,20)(21,24)(22,25)(23,31)(26,28)(27,32)(29,34)(30,36)(33,35);(5,9,10,18)(6,14,11,19)(7,15,13,20)(8,16,12,17)(21,25)(22,28)(23,32)(24,29)(26,34)(27,35)(30,31)(33,36)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=h,f^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, 7562469);