#561979

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2106 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fgh,e^2=fg,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, 469175);