#371303

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1359 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278499);