#6439304

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23391	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2455 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)(15,19)(16,24)(17,21)(18,22)(20,23)(25,27)(26,28);(1,2)(3,4)(6,9)(10,12)(13,15,18,17)(14,19,22,21)(16,23,25,26)(20,27,28,24);(2,4)(13,16)(14,20)(15,23)(17,26)(18,25)(19,27)(21,24)(22,28);(2,4)(5,7,8,11)(6,10,9,12);(13,15,18,17)(14,21,22,19)(16,23,25,26)(20,24,28,27)

Polycyclic presentation:

2^9|b^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,d^b=dh,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307335, #72307422, #72307437
maximal normal subgroup of	#72307335, #72307422, #72307437

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346500);