#444432

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

782 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)(5,6,9,11)(7,10,8,12)(13,14,18,21)(15,22,26,19)(16,24,20,17)(23,27,28,25);(5,7,9,8)(6,10,11,12)(14,16)(15,23)(17,24)(19,22)(20,21)(26,28);(6,11)(10,12)(13,15,24,28,18,26,17,23)(14,19,20,27,21,22,16,25);(2,4)(5,8,9,7)(6,12,11,10)(13,16,18,20)(14,17,21,24)(15,25,26,27)(19,23,22,28)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351628);