#455969

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
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	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1182 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)(3,4)(6,9)(10,12)(13,14)(15,25)(16,19)(17,27)(18,23)(20,24)(21,26)(22,28);(5,6,8,9)(7,10,11,12)(13,15)(14,21)(16,17)(18,22)(19,26)(20,27)(23,25)(24,28);(5,7,8,11)(6,10,9,12)(13,16,18,24,20,23,19,14)(15,25,26,28,27,17,22,21);(2,4)(7,11)(10,12)(13,17,20,25)(14,22,24,26)(15,16,27,23)(18,28,19,21)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363165);