#557226

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fg,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 464422);