#452791

order	512
nilpotency class	3
isoclinism	isoclinic class of #418488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1031 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

2^9|a^2=egh,b^2=efg,c^2=ef,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^b=hi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339668, #72339678, #72339683
maximal normal subgroup of	#72339668, #72339678, #72339683

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359987);