#112918

order	512
nilpotency class	3
isoclinism	isoclinic class of #112912
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#35292	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,g^b=ghi,e^c=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316676, #72316769, #72316773
maximal normal subgroup of	#72316676, #72316769, #72316773

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20114);