#582222

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3048 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=g,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293260
maximal normal subgroup of	#72293260

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 489418);