#6578878

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

2168 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)(7,11)(8,10)(9,12)(14,20)(15,24)(22,28)(23,27);(5,7,9,8)(6,10,12,11)(13,14,18,20)(15,25,24,26)(16,22,21,28)(17,23,19,27);(2,4)(5,7,9,8)(6,11,12,10)(13,15,18,24)(14,21,20,16)(17,22,19,28)(23,26,27,25);(5,7,9,8)(6,11,12,10)(13,16)(14,22)(15,23)(17,26)(18,21)(19,25)(20,28)(24,27);(5,8,9,7)(6,10,12,11)(13,17)(15,22)(18,19)(23,27)(24,28)(25,26)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=fi,e^2=fi,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	#72307082, #72307099, #72307632
maximal normal subgroup of	#72307082, #72307099, #72307632

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486074);