#2316232

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

2081 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)(6,9)(7,8)(21,22,25,26)(23,27,29,33)(24,28,32,34)(30,35,31,36);(5,6,10,9)(7,12,8,11)(13,14,16,18)(15,17,19,20)(22,26)(23,29)(28,34)(30,31);(5,7,10,8)(6,11,9,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,27)(24,31)(25,29)(26,33)(28,36)(30,32)(34,35);(1,2,3,4)(5,8,10,7)(6,11,9,12)(21,22)(23,27)(24,28)(25,26)(29,33)(30,35)(31,36)(32,34);(5,9,10,6)(7,11,8,12)(13,15,16,19)(14,17,18,20)(21,24,25,32)(22,28,26,34)(23,30,29,31)(27,35,33,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=g,d^2=fgi,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, 2223428);