#7626789

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×#3455	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6654 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)(5,6,12,15)(7,16,10,13)(8,11,9,14)(17,18,19,20)(21,22,28,31)(23,26,30,36)(24,33,35,29)(25,27,34,32);(5,7,12,10)(6,13,15,16)(8,19,9,17)(11,18,14,20)(21,23,28,30)(22,26,31,36)(24,32,35,27)(25,29,34,33);(1,3,4,2)(5,8,12,9)(6,14,15,11)(7,17,10,19)(13,18,16,20)(21,24,28,35)(22,29,31,33)(23,32,30,27)(25,26,34,36);(5,9,12,8)(6,11,15,14)(7,17,10,19)(13,18,16,20)(21,25)(22,27)(23,29)(24,26)(28,34)(30,33)(31,32)(35,36);(5,10,12,7)(6,13,15,16)(8,19,9,17)(11,20,14,18)(21,26)(22,30)(23,31)(24,34)(25,35)(27,29)(28,36)(32,33);(5,11)(6,8)(7,18)(9,15)(10,20)(12,14)(13,19)(16,17)(21,27)(22,25)(23,24)(26,29)(28,32)(30,35)(31,34)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 7533985);