#6625600

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3007 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284628, #72285443
maximal normal subgroup of	#72284628, #72285443

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6532796);