#561049

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2110 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=f,e^2=fh,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, 468245);