#392265

order	512
nilpotency class	3
isoclinism	isoclinic class of #32888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3542 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)(7,10)(8,9)(11,12)(13,14,17,21,18,16,25,20)(15,19,23,27,24,22,28,26);(6,9)(10,12)(13,15,18,24)(14,19,16,22)(17,23,25,28)(20,26,21,27);(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,22)(17,20)(19,24)(21,25)(23,26)(27,28);(2,4)(13,14)(15,19)(16,18)(17,20)(21,25)(22,24)(23,26)(27,28)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,f^c=fi,g^d=gi,h^d=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299461);