#504703

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	#13325	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

625 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)(7,12)(8,10)(9,11)(14,15)(16,17)(18,26)(20,27)(21,24)(22,28)(23,25);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,14,17,21,18,22,27,25,19,23,20,28,26,24,16,15);(2,4)(5,8,9,7)(6,10,11,12)(13,15,16,24,26,28,20,23,19,25,27,22,18,21,17,14);(6,11)(10,12)(13,14,20,28,18,22,16,15,19,23,17,21,26,24,27,25)

Polycyclic presentation:

2^9|b^2=eh,c^2=e,d^2=fi,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^b=di,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319014, #72319524, #72319591
maximal normal subgroup of	#72319014, #72319524, #72319591
maximal quotient of	#72322603

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411899);