#7626325

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	C₂×#121	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6438 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)(5,6,11,16)(7,14,9,12)(8,18,19,13)(10,15,17,20)(21,22,27,24)(23,26,31,28)(25,29,33,32)(30,34,36,35);(5,7,11,9)(6,12,16,14)(8,10,19,17)(13,15,18,20)(25,33)(29,32)(30,36)(34,35);(5,8,11,19)(6,13,16,18)(7,17,9,10)(12,20,14,15)(21,23,27,31)(22,28,24,26)(25,30,33,36)(29,35,32,34);(1,2,4,3)(5,9,11,7)(6,14,16,12)(8,10,19,17)(13,15,18,20)(21,24,27,22)(23,26,31,28)(25,32,33,29)(30,34,36,35);(1,3,4,2)(5,10)(6,15)(7,8)(9,19)(11,17)(12,13)(14,18)(16,20)(21,25,27,33)(22,29,24,32)(23,30,31,36)(26,34,28,35);(5,6)(7,12)(8,13)(9,14)(10,15)(11,16)(17,20)(18,19)(21,26)(22,23)(24,31)(25,34)(27,28)(29,30)(32,36)(33,35)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=i,e^2=gi,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, 7533521);