#6466161

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2344 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373357);