#557392

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3386 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)(8,9)(10,12)(13,14)(15,19)(16,20)(17,18)(21,22,25,26)(23,27,28,24);(6,9)(7,12)(14,18)(16,19)(21,23,25,28)(22,24,26,27);(1,3,4,2)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,27)(26,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23)(25,27)(26,28)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213086
maximal normal subgroup of	#72213086
maximal quotient of	#72267454, #72267612, #72267682

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 464588);