#6576543

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

2883 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)(14,19)(15,22)(21,27)(24,28);(5,7,9,8)(6,10,11,12)(13,14,17,19)(15,23,22,25)(16,24,20,28)(18,27,26,21);(2,4)(13,15)(14,20)(16,19)(17,22)(18,28)(21,25)(23,27)(24,26);(5,8,9,7)(6,10,11,12)(15,24)(16,25)(18,26)(20,23)(21,27)(22,28);(13,16,17,20)(14,15,19,22)(18,23,26,25)(21,28,27,24)

Polycyclic presentation:

2^9|b^2=f,d^2=fi,e^2=i,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	#72307099, #72307108, #72307789
maximal normal subgroup of	#72307099, #72307108, #72307789

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483739);