#562468

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

1842 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fg,d^2=fh,e^2=gh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 469664);