#6432030

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3704 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)(7,10)(8,9)(11,12)(13,14,18,22)(15,23,25,28)(16,21,17,20)(19,26,27,24);(1,2)(3,4)(6,9)(10,12);(2,4)(13,15)(14,19)(16,24)(17,26)(18,25)(20,23)(21,28)(22,27);(5,7,8,11)(6,10,9,12);(13,16,18,17)(14,20,22,21)(15,24,25,26)(19,23,27,28)

Polycyclic presentation:

2^9|a^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315268, #72315279, #72315312
maximal normal subgroup of	#72315268, #72315279, #72315312
maximal quotient of	#72316468, #72342008

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339226);