#226957

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	C₂×#453	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

802 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dei,e^b=eh,d^c=dh,f^c=fh,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215940
maximal normal subgroup of	#72215940
maximal quotient of	#72215902

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134153);