#474778

order	512
nilpotency class	3
isoclinism	isoclinic class of #469776
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
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

1036 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339676, #72339686, #72339765
maximal normal subgroup of	#72339676, #72339686, #72339765

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381974);