#7626940

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×#535	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6628 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)(5,6,12,15)(7,16,10,13)(8,11,9,14)(17,18,19,20)(21,22)(23,24)(25,33)(26,29)(27,31)(28,34)(30,35)(32,36);(5,7,12,10)(6,13,15,16)(8,19,9,17)(11,18,14,20);(5,8,12,9)(6,14,15,11)(7,17,10,19)(13,18,16,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,35)(32,33)(34,36);(5,9,12,8)(6,11,15,14)(7,17,10,19)(13,18,16,20)(21,24,26,27)(22,23,29,31)(25,32,34,35)(28,30,33,36);(1,3,4,2)(5,10,12,7)(6,13,15,16)(8,19,9,17)(11,20,14,18)(21,23)(22,24)(25,30)(26,31)(27,29)(28,32)(33,35)(34,36);(5,11)(6,8)(7,18)(9,15)(10,20)(12,14)(13,19)(16,17)(21,25,26,34)(22,28,29,33)(23,30,31,36)(24,32,27,35)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=h,e^2=i,f^2=gh,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, 7534136);