#432466

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

976 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)(6,9)(7,11)(13,14,19,20,18,22,17,16)(15,23,24,25,27,21,26,28);(5,6,8,9)(7,10,11,12)(14,20)(15,24)(16,22)(17,19)(21,23)(26,27);(5,7,8,11)(6,10,9,12)(13,15,18,27)(14,21,22,23)(16,25,20,28)(17,26,19,24);(2,4)(6,9)(10,12)(13,16,17,22,18,20,19,14)(15,25,26,23,27,28,24,21)

Polycyclic presentation:

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

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, 339662);