#244853

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
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
prime radical	C₂×#3284	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1328 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)(6,10)(7,8)(11,12)(13,14)(15,24)(16,20)(17,21)(18,23)(19,25)(22,27)(26,28);(6,11)(7,8)(10,12)(13,15,18,16)(14,19,23,28)(17,26,27,25)(20,21,24,22);(2,4)(5,6,9,12)(7,11,8,10)(13,16,18,15)(14,20,23,24)(17,26,27,25)(19,21,28,22);(14,21)(17,27)(19,24)(20,28)(22,23)(25,26)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,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

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310301, #72310303, #72310313
maximal normal subgroup of	#72310301, #72310303, #72310313

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152049);