#544790

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

2526 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)(6,9)(10,12)(14,18)(16,19)(22,27)(23,25);(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,19,20,16)(21,22,26,27)(23,24,25,28);(7,11)(10,12)(14,18)(15,20)(21,23,26,25)(22,28,27,24);(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,18)(17,20)(21,24,26,28)(22,23,27,25);(5,6)(7,10)(8,9)(11,12)(13,16,17,19)(14,15,18,20)(21,25,26,23)(22,28,27,24)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=g,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,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, 451986);