#6682845

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2130 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=gi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256945, #72257043, #72257071
maximal normal subgroup of	#72256945, #72257043, #72257071
maximal quotient of	#72264422, #72264527

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590041);