#6667554

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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

2941 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)(13,14,17,18)(15,19,23,21)(16,20,24,25)(22,26,28,27);(5,7)(6,9)(8,12)(10,11)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27);(2,4)(14,18)(15,21)(19,23)(20,25)(22,27)(26,28);(6,10)(9,11)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27);(7,12)(9,11)(13,14)(16,20)(17,18)(19,21)(24,25)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283019, #72283104, #72284455
maximal normal subgroup of	#72283019, #72283104, #72284455

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6574750);