#554157

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
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

1992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

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, 461353);