#509872

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=ei,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343483, #72343811, #72343857
maximal normal subgroup of	#72343483, #72343811, #72343857

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417068);