#346415

order	512
nilpotency class	3
isoclinism	isoclinic class of #346398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3536	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253611);