#2280928

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
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

2906 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fhi,c^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333438, #72333462, #72333479
maximal normal subgroup of	#72333438, #72333462, #72333479

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2188124);