#536071

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

2790 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=g,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, 443267);