#576194

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁴×C₄³

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

2670 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=fg,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, 483390);