#7627134

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
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

6370 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)(7,14)(8,11)(9,12)(10,16)(13,18)(15,19)(17,20)(21,22,27,25)(23,30,26,28)(24,29,33,34)(31,36,32,35);(1,2,4,3)(5,7,11,9)(6,12,8,14)(10,15,18,17)(13,19,16,20);(5,6,11,8)(7,14,9,12)(10,13,18,16)(15,20,17,19)(21,23,27,26)(22,28,25,30)(24,31,33,32)(29,35,34,36);(5,8,11,6)(7,14,9,12)(10,16,18,13)(15,20,17,19)(21,24)(22,29)(23,31)(25,34)(26,32)(27,33)(28,35)(30,36);(1,3,4,2)(5,9,11,7)(6,12,8,14)(10,17,18,15)(13,19,16,20)(21,25,27,22)(23,30,26,28)(24,29,33,34)(31,35,32,36);(5,10,11,18)(6,13,8,16)(7,15,9,17)(12,19,14,20)(21,26,27,23)(22,30,25,28)(24,32,33,31)(29,36,34,35)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534330);