#535768

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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

2570 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)(5,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,18,23,24)(19,22,21,20);(9,10,13,14)(11,12,16,15)(17,19,23,21)(18,20,24,22);(1,2,4,3)(10,14)(11,16)(17,20,23,22)(18,19,24,21);(1,3,4,2)(9,11)(10,12)(13,16)(14,15)(17,21,23,19)(18,20,24,22);(9,12)(10,11)(13,15)(14,16)(17,22,23,20)(18,19,24,21)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fg,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 442964);