#7626733

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	C₂²×#41	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6434 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)(7,11)(8,16)(9,13)(10,15)(12,17)(14,18)(19,20)(21,22)(23,28)(24,26)(25,32)(27,31)(29,33)(30,34)(35,36);(1,3,4,2)(21,23)(22,27)(24,28)(25,30)(26,31)(29,34)(32,35)(33,36);(5,7)(6,10)(8,14)(9,15)(11,13)(12,18)(16,19)(17,20)(21,22,26,24)(23,28,31,27)(25,29,33,32)(30,35,36,34);(5,7)(6,11)(8,14)(9,15)(10,13)(12,19)(16,18)(17,20)(21,24,26,22)(23,28,31,27)(25,32,33,29)(30,35,36,34);(5,6)(7,10)(8,12)(9,13)(11,15)(14,18)(16,17)(19,20)(21,23)(22,28)(24,27)(25,30)(26,31)(29,35)(32,34)(33,36);(5,8,9,17)(6,12,13,16)(7,14,15,20)(10,18,11,19)(21,25,26,33)(22,29,24,32)(23,30,31,36)(27,34,28,35)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=g,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, 7533929);