#517898

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2768 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fgh,d^2=i,e^2=fgh,b^a=bf,c^a=cg,c^b=ch,e^d=ef

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 425094);