#335073

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25759	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

873 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)(3,4)(5,6,10,11,9,7,8,12)(14,19)(15,23)(21,22)(25,26);(5,7,9,6)(8,12,10,11)(13,14,17,19)(15,24,23,28)(16,22,18,21)(20,26,27,25);(2,4)(6,7)(11,12)(13,15,17,23)(14,20,19,27)(16,25,18,26)(21,24,22,28);(2,4)(6,11)(7,12)(8,10)(13,16,17,18)(14,21,19,22)(15,25,23,26)(20,24,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 242269);