#433266

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1203 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2,4,3)(6,10)(11,12)(13,14,19,22,17,16,18,21)(15,23,27,28,25,20,26,24);(2,3)(5,6)(7,11)(8,12)(9,10)(13,15,18,26,17,25,19,27)(14,20,21,28,16,23,22,24);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,17,14)(15,24,25,28)(18,21,19,22)(20,26,23,27);(1,3,4,2)(5,8,9,7)(6,11,10,12)(13,14,17,16)(15,20,25,23)(18,21,19,22)(24,26,28,27)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,f^c=fi,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226919, #72226925, #72226941
maximal normal subgroup of	#72226919, #72226925, #72226941

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340462);