#287744

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
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	#28369	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

831 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)(13,14,18,22,17,21,24,27)(15,23,25,20,16,19,26,28);(1,2,4,3)(5,6)(7,9)(8,11)(10,12)(13,15,24,26,17,16,18,25)(14,19,27,20,21,23,22,28);(6,9)(7,12)(10,11)(13,16,17,15)(14,20,21,28)(18,25,24,26)(19,22,23,27);(1,3,4,2)(14,21)(19,23)(20,28)(22,27)

Polycyclic presentation:

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

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, 194940);