#7627417

order	512
nilpotency class	2
isoclinism	isoclinic class of #63041
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×#3531	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6294 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh

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, 7534613);