#2329214

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=gh,d^2=hi,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, 2236410);