#7627455

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×#121	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6206 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)(7,8)(9,13)(10,14)(11,15)(12,18)(16,17)(19,20)(21,22)(23,31)(24,25)(26,28)(27,30)(29,36)(32,33)(34,35);(5,7)(6,8)(9,10)(11,12)(13,14)(15,18)(16,19)(17,20)(21,23,27,26)(22,28,30,31)(24,34,29,32)(25,33,36,35);(5,8)(6,12)(7,15)(9,17)(10,13)(11,18)(14,19)(16,20)(21,24,27,29)(22,25,30,36)(23,32,26,34)(28,35,31,33);(5,9)(6,13)(7,16)(8,17)(10,12)(11,19)(14,18)(15,20)(21,25)(22,29)(23,33)(24,30)(26,35)(27,36)(28,32)(31,34);(1,3,4,2)(5,8)(6,7)(9,17)(10,20)(11,18)(12,15)(13,16)(14,19)(21,26,27,23)(22,28,30,31)(24,34,29,32)(25,35,36,33);(5,10,11,16)(6,14,15,17)(7,9,12,19)(8,13,18,20)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,e^2=gi,f^2=h,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, 7534651);