#429018

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1043 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)(13,14,20,17,19,15,18,22)(16,23,27,28,26,21,25,24);(5,6)(7,10)(8,9)(11,12)(13,15,19,14)(16,23,26,21)(17,18,22,20)(24,25,28,27);(5,7,8,11)(6,10,9,12)(13,16,18,25,19,26,20,27)(14,21,22,28,15,23,17,24);(2,4)(6,9)(10,12)(13,17,18,14,19,22,20,15)(16,24,25,21,26,28,27,23)

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=e,d^2=efi,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,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, 336214);