#1100290

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2858 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:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=fg,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326419, #72326420, #72326446
maximal normal subgroup of	#72326419, #72326420, #72326446

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1007486);