#249665

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1355 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)(5,6,8,11)(7,9,12,10)(13,14,19,22)(15,17,21,18)(16,24,28,23)(20,26,25,27);(6,9)(7,12)(10,11)(13,15)(14,17)(16,25)(18,22)(19,21)(20,28)(23,26)(24,27);(2,4)(13,16,19,28)(14,20,22,25)(15,23,21,24)(17,26,18,27);(6,10)(7,12)(9,11)(13,17,19,18)(14,21,22,15)(16,26,28,27)(20,24,25,23)

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=deh,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310301, #72310306, #72310313
maximal normal subgroup of	#72310301, #72310306, #72310313

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156861);