#441078

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35528	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1821 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:

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

Polycyclic presentation:

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

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, 348274);