#6491008

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470122
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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

3584 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)(7,11)(10,12)(13,14)(15,23)(16,19)(17,21)(18,22)(20,25)(24,27)(26,28);(2,4)(5,6)(7,10)(8,9)(11,12)(14,19)(15,24)(16,17)(20,23)(21,22)(27,28);(6,9)(10,12)(13,15,18,24)(14,20,22,28)(16,25,17,26)(19,27,21,23);(6,9)(10,12)(13,16,18,17)(14,21,22,19)(15,25,24,26)(20,23,28,27);(5,7,8,11)(6,10,9,12)(14,22)(19,21)(20,28)(23,27)

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72337463, #72337549
maximal normal subgroup of	#72337463, #72337549

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398204);