#580920

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
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	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3076 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fh,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333364, #72333365, #72333428
maximal normal subgroup of	#72333364, #72333365, #72333428

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 488116);