#7626213

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	C₂⁶	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,e^2=g,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, 7533409);