#6576556

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2890 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)(7,12)(8,10)(9,11)(13,14,18,21)(15,16,19,26)(17,20,27,28)(22,24,23,25);(5,7,9,8)(6,10,11,12)(13,15,18,19)(14,16,21,26)(17,22,27,23)(20,24,28,25);(2,4)(15,22)(16,24)(17,27)(19,23)(20,28)(25,26);(5,8,9,7)(6,10,11,12)(13,16)(14,19)(15,21)(17,24)(18,26)(20,23)(22,28)(25,27);(15,22)(16,24)(17,27)(19,23)(20,28)(25,26)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307067, #72307195, #72307399
maximal normal subgroup of	#72307067, #72307195, #72307399

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483752);