#7626518

order	512
nilpotency class	2
isoclinism	isoclinic class of #62862
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×#3566	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533714);