#6470088

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470077
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 ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4449 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282681, #72282778, #72282979
maximal normal subgroup of	#72282681, #72282778, #72282979

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6377284);