#283031

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

796 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fh,c^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gi,d^c=di,g^c=gi,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 190227);