#7627522

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×#3515	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6382 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=g,e^2=g,f^2=g,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, 7534718);