#500319

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

814 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319285, #72319586, #72319645, …
maximal normal subgroup of	#72319285, #72319586, #72319645, …
maximal quotient of	#72322632, #72322825

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407515);