#7626136

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator	C₂²	C₂⁵×C₄
Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3482	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6710 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)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19)(21,22)(23,29)(24,32)(25,28)(26,31)(27,34)(30,35)(33,36);(5,7,10,17)(6,11,13,15)(8,18,9,16)(12,20,14,19)(21,23,25,31)(22,26,28,29)(24,33,34,30)(27,36,32,35);(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,24,25,34)(22,27,28,32)(23,30,31,33)(26,35,29,36);(1,3,4,2)(6,13)(11,15)(12,14)(19,20)(22,28)(23,31)(27,32)(30,33);(22,28)(26,29)(27,32)(35,36);(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)

Polycyclic presentation:

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

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, 7533332);