#7626861

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

6458 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)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,6,10,9)(7,11,15,13)(8,12,16,17)(14,18,20,19)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(5,7)(6,11)(8,14)(9,13)(10,15)(12,18)(16,20)(17,19)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(5,8)(6,12)(7,14)(9,17)(10,16)(11,18)(13,19)(15,20)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(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)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,e^2=i,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, 7534057);