#6466554

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,11)(8,9)(10,12)(13,14,19,24)(15,25,28,41)(16,21,17,22)(18,32,33,23)(20,34,36,27)(26,40,29,43)(30,39,31,38)(35,44,37,42);(2,3)(6,9)(10,11)(13,15,18,27,19,28,33,34)(14,20,23,25,24,36,32,41)(16,29,30,44,17,26,31,42)(21,37,38,43,22,35,39,40);(1,2,4,3)(5,7,8,12)(6,10,9,11)(13,16,19,17)(14,21,24,22)(15,26,28,29)(18,30,33,31)(20,35,36,37)(23,38,32,39)(25,40,41,43)(27,42,34,44);(1,3,4,2)(13,17,19,16)(14,22,24,21)(15,26,28,29)(18,31,33,30)(20,35,36,37)(23,39,32,38)(25,40,41,43)(27,42,34,44);(16,17)(21,22)(26,29)(30,31)(35,37)(38,39)(40,43)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=gh,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373750);