#574185

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

1978 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)(5,6)(7,11)(8,10)(9,12)(13,14,17,19)(15,16,20,18)(21,22)(23,28)(24,30)(25,29)(26,27)(31,35)(32,36)(33,34);(1,2,4,3)(5,7,10,12)(6,9,8,11)(13,15)(14,18)(16,19)(17,20)(22,27)(23,30)(25,33)(32,35);(6,8)(7,12)(13,16,17,18)(14,15,19,20)(21,23,26,30)(22,28,27,24)(25,31,33,36)(29,35,34,32);(1,2,4,3)(5,8,10,6)(7,9,12,11)(13,16)(14,15)(17,18)(19,20)(21,24,26,28)(22,23,27,30)(25,32,33,35)(29,31,34,36);(1,3,4,2)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,25)(22,29)(23,31)(24,32)(26,33)(27,34)(28,35)(30,36)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=fi,e^2=fhi,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, 481381);