#287828

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

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

715 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)(5,6,9,7,10,12,8,11)(13,14,18,21,17,22,16,20)(15,23,26,28,24,19,25,27);(1,2,4,3)(5,7,8,6,10,11,9,12)(13,15,16,25,17,24,18,26)(14,19,20,28,22,23,21,27);(6,7)(8,9)(11,12)(14,20)(15,24)(16,18)(19,27)(21,22)(23,28);(1,3,4,2)(13,16,17,18)(14,21,22,20)(15,25,24,26)(19,27,23,28)

Polycyclic presentation:

2^9|a^2=fhi,b^2=ef,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, 195024);