#510008

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	C₂×#2206	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

643 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,c^2=e,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	#72343459, #72343826, #72343846
maximal normal subgroup of	#72343459, #72343826, #72343846

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417204);