#532804

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2806 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fgi,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=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, 440000);