#348748

order	512
nilpotency class	3
isoclinism	isoclinic class of #346582
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1171 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=i,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,f^c=fh,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255944);