#6467644

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25794	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2380 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)(7,10)(8,9)(11,12)(13,14,19,16,20,24,18,23)(15,17,22,25,26,27,28,21);(2,4)(13,15)(14,21)(16,27)(17,23)(18,22)(19,28)(20,26)(24,25);(6,9)(10,12)(13,16,19,24,20,23,18,14)(15,25,22,27,26,21,28,17);(13,17,18,21,20,27,19,25)(14,22,23,15,24,28,16,26);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,24,16)(15,22,26,28)(17,25,27,21)

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=hi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374840);