#502009

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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×#2215	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

616 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)(7,9)(8,10)(11,12)(13,14)(15,24)(16,26)(17,27)(18,28)(19,23)(20,22)(21,25);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,25,23,18,16,26,28,19,21,24,14,20,27,17,22);(2,4)(6,10)(9,12);(13,16,25,28,18,21,26,14,19,27,24,22,20,15,17,23)

Polycyclic presentation:

2^9|b^2=efh,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319014, #72319560, #72319567
maximal normal subgroup of	#72319014, #72319560, #72319567
maximal quotient of	#72322616

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409205);