#405110

order	512
nilpotency class	3
isoclinism	isoclinic class of #393877
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3006	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=hi,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302575, #72302843, #72302850
maximal normal subgroup of	#72302575, #72302843, #72302850

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312306);