#221361

order	512
nilpotency class	3
isoclinism	isoclinic class of #221164
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61496	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

953 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,e^2=h,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,d^b=de,e^b=eh,f^c=fhi,g^c=gh,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128557);