#553801

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

1966 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)(6,7)(10,11)(13,14)(15,17)(16,18)(19,20);(5,6,9,7)(8,11,12,10)(14,15)(16,20)(21,22,24,26)(23,27,28,25);(13,14,17,15)(16,18,20,19)(21,23,24,28)(22,25,26,27);(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18);(5,8)(6,10)(7,11)(9,12)(13,16,17,20)(14,18,15,19)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=h,d^2=f,e^2=g,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, 460997);