#504755

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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

660 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343301, #72343829, #72343845
maximal normal subgroup of	#72343301, #72343829, #72343845

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411951);