#553953

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

1862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=g,e^2=fgh,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, 461149);