#572265

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

1880 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)(5,6,8,10)(7,9,11,12)(21,22,26,27)(23,25,31,28)(24,32,33,29)(30,36,34,35);(7,11)(9,12)(13,14,17,16)(15,19,20,18)(22,27)(24,33)(25,28)(30,34);(5,6)(7,9)(8,10)(11,12)(13,15,17,20)(14,18,16,19)(21,23)(22,28)(24,34)(25,27)(26,31)(29,35)(30,33)(32,36);(1,2,4,3)(5,7,8,11)(6,9,10,12)(21,24,26,33)(22,29,27,32)(23,30,31,34)(25,35,28,36);(1,3,4,2)(5,6)(7,9)(8,10)(11,12)(13,16,17,14)(15,19,20,18)(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=h,c^2=h,d^2=i,e^2=fghi,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, 479461);