#7627062

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

6288 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)(5,6,10,8)(7,13,15,11)(9,12,17,16)(14,19,20,18)(21,22,27,26)(23,30,25,28)(24,29,34,33)(31,36,32,35);(1,2,4,3)(21,23,27,25)(22,28,26,30)(24,32,34,31)(29,36,33,35);(5,7)(6,11)(8,13)(9,14)(10,15)(12,18)(16,19)(17,20)(21,24,27,34)(22,29,26,33)(23,31,25,32)(28,35,30,36);(5,8,10,6)(7,13,15,11)(9,12,17,16)(14,18,20,19)(21,25,27,23)(22,30,26,28)(24,32,34,31)(29,36,33,35);(1,3,4,2)(5,9,10,17)(6,12,8,16)(7,14,15,20)(11,18,13,19);(5,7)(6,11)(8,13)(9,14)(10,15)(12,18)(16,19)(17,20)(21,26,27,22)(23,30,25,28)(24,33,34,29)(31,36,32,35)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=gh,e^2=gi,f^2=g,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, 7534258);