#429980

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23430	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1818 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 337176);