#7626868

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×#525	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,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, 7534064);