#6452210

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

3594 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305348, #72305349, #72305376
maximal normal subgroup of	#72305348, #72305349, #72305376
maximal quotient of	#72305553, #72305655

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6359406);