#7626790

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×#537	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6472 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)(5,6,12,15)(7,10,14,20)(8,17,19,13)(9,11,18,16)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,7,12,14)(6,10,15,20)(8,16,19,11)(9,13,18,17)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(1,3,4,2)(5,8,12,19)(6,13,15,17)(7,16,14,11)(9,10,18,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(5,9)(6,11)(7,13)(8,10)(12,18)(14,17)(15,16)(19,20)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35);(5,10)(6,14)(7,15)(8,18)(9,19)(11,13)(12,20)(16,17)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35);(5,11)(6,9)(7,8)(10,13)(12,16)(14,19)(15,18)(17,20)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=g,e^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, 7533986);