#398922

order	512
nilpotency class	3
isoclinism	isoclinic class of #395148
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

967 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=fi,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306118);