#2329545

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=fgh,d^2=fghi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2236741);