#367718

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

987 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)(5,6)(7,10)(8,9)(11,12)(13,14,18,22,19,23,20,24)(15,25,21,16,26,27,28,17);(1,3,4,2)(13,15,18,28,19,26,20,21)(14,17,22,27,23,16,24,25);(6,9)(10,12)(13,16)(14,15)(17,19)(18,27)(20,25)(21,22)(23,26)(24,28);(5,7,8,11)(6,10,9,12)(13,17)(14,21)(15,22)(16,19)(18,27)(20,25)(23,28)(24,26)

Polycyclic presentation:

2^9|a^2=h,b^2=egh,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,e^b=ei,h^b=hi,g^c=gi,e^d=ei,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, 274914);