#582304

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1876 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=h,e^2=fg,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 489500);