#7626394

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×#3506	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6888 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)(7,11)(8,15)(9,10)(12,16)(13,17)(14,20)(18,19)(21,22)(23,26)(24,27)(25,30)(28,33)(29,32)(31,35)(34,36);(7,13)(11,17)(14,19)(18,20)(21,22,26,23)(24,29,32,27)(25,28,33,30)(31,35,36,34);(1,3,4,2)(6,10)(11,17)(12,15)(18,20)(22,23)(27,29)(28,30)(34,35);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,23,26,22)(24,29,32,27)(25,30,33,28)(31,35,36,34);(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,20)(16,19)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,8)(6,12)(7,14)(9,16)(10,15)(11,18)(13,19)(17,20)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,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^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, 7533590);