#148979

order	512
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#476
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=d,d^2=h,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214451, #72215349, #72216354, …
maximal normal subgroup of	#72214451, #72215349, #72216354, …
maximal quotient of	#72214231

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56175);