#6452082

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3510 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)(7,12)(8,10)(9,11)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,27)(25,28);(1,2)(3,4)(13,15,17,24,18,25,16,23)(14,19,20,22,21,27,26,28);(2,4)(5,7,9,8)(6,10,11,12)(13,14)(15,19)(16,26)(17,20)(18,21)(22,24)(23,28)(25,27);(2,4)(5,8,9,7)(6,10,11,12);(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305353, #72305359, #72305368
maximal normal subgroup of	#72305353, #72305359, #72305368
maximal quotient of	#72305564, #72305572

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6359278);