#513958

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂²	C₂³×C₄²
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3526 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 421154);