#7626341

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
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×#3553	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6530 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)(6,11)(9,17)(12,18)(16,20)(22,24)(23,31)(26,36)(29,33);(8,15)(12,18)(14,19)(16,20)(21,22,27,24)(23,32,31,28)(25,33,35,29)(26,30,36,34);(5,6,10,11)(7,9,13,17)(8,12,15,18)(14,16,19,20)(21,23)(22,28)(24,32)(25,26)(27,31)(29,30)(33,34)(35,36);(1,2,4,3)(5,7,10,13)(6,9,11,17)(8,14,15,19)(12,16,18,20)(21,24,27,22)(23,32,31,28)(25,33,35,29)(26,34,36,30);(1,3,4,2)(5,8)(6,12)(7,14)(9,16)(10,15)(11,18)(13,19)(17,20)(21,25)(22,29)(23,26)(24,33)(27,35)(28,30)(31,36)(32,34);(5,9)(6,13)(7,11)(8,16)(10,17)(12,19)(14,18)(15,20)(21,26)(22,30)(23,25)(24,34)(27,36)(28,29)(31,35)(32,33)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=hi,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533537);