#7626275

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	C₂⁴	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6418 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)(7,14,9,12)(8,18,17,13)(10,15,19,20)(21,22,27,25)(23,28,24,31)(26,33,34,29)(30,36,32,35);(5,7,11,9)(6,12,16,14)(8,19,17,10)(13,20,18,15)(21,23,27,24)(22,28,25,31)(26,30,34,32)(29,35,33,36);(1,3,4,2)(5,8)(6,13)(7,10)(9,19)(11,17)(12,15)(14,20)(16,18)(21,24,27,23)(22,28,25,31)(26,32,34,30)(29,35,33,36);(5,9,11,7)(6,14,16,12)(8,19,17,10)(13,20,18,15);(5,10,11,19)(6,15,16,20)(7,17,9,8)(12,18,14,13)(21,25,27,22)(23,28,24,31)(26,33,34,29)(30,35,32,36);(5,6)(7,12)(8,13)(9,14)(10,15)(11,16)(17,18)(19,20)(21,26)(22,29)(23,30)(24,32)(25,33)(27,34)(28,35)(31,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=g,e^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, 7533471);