#7626308

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3512	C₂
LCS	G ▷ C₂² ▷ 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533504);