#400122

order	512
nilpotency class	3
isoclinism	isoclinic class of #394048
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1841 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:

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

Polycyclic presentation:

2^9|a^2=ei,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307318);