#505202

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

808 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319014, #72319542, #72319654
maximal normal subgroup of	#72319014, #72319542, #72319654
maximal quotient of	#72322655

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412398);