#7626565

order	512
nilpotency class	2
isoclinism	isoclinic class of #62862
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×#3544	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6452 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)(5,6,11,8)(7,14,9,12)(10,16,18,13)(15,19,17,20)(21,22,26,23)(24,29,32,27)(25,28,33,30)(31,35,36,34);(5,6,11,8)(7,12,9,14)(10,13,18,16)(15,19,17,20)(22,23)(27,29)(28,30)(34,35);(5,7,11,9)(6,12,8,14)(10,15,18,17)(13,19,16,20)(21,23,26,22)(24,29,32,27)(25,30,33,28)(31,35,36,34);(5,8,11,6)(7,12,9,14)(10,16,18,13)(15,19,17,20)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(1,3,4,2)(5,9,11,7)(6,12,8,14)(10,17,18,15)(13,19,16,20)(21,25,26,33)(22,28,23,30)(24,31,32,36)(27,34,29,35);(5,10,11,18)(6,13,8,16)(7,15,9,17)(12,19,14,20)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533761);