#7627135

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×#3544	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6598 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)(5,6,10,13)(7,8,15,11)(9,17,18,12)(14,19,20,16)(21,22,27,25)(23,30,26,28)(24,29,33,34)(31,36,32,35);(1,2,4,3)(5,6)(7,8)(9,12)(10,13)(11,15)(14,16)(17,18)(19,20);(5,7)(6,11)(8,13)(9,14)(10,15)(12,19)(16,17)(18,20)(21,23,27,26)(22,28,25,30)(24,31,33,32)(29,35,34,36);(5,8,10,11)(6,7,13,15)(9,16,18,19)(12,14,17,20)(21,24)(22,29)(23,31)(25,34)(26,32)(27,33)(28,35)(30,36);(1,3,4,2)(5,7)(6,8)(9,14)(10,15)(11,13)(12,16)(17,19)(18,20)(21,25,27,22)(23,30,26,28)(24,29,33,34)(31,35,32,36);(5,9,10,18)(6,12,13,17)(7,14,15,20)(8,16,11,19)(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=gh,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, 7534331);