#287843

order	512
nilpotency class	3
isoclinism	isoclinic class of #281659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

730 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=i,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 195039);