#196393

order	512
nilpotency class	3
isoclinism	isoclinic class of #162406
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1344 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)(6,10,11,12)(14,18)(16,19)(20,24)(21,25)(23,26)(27,28);(5,6,8,12,9,11,7,10)(13,14,19,25,17,21,16,18)(15,20,26,28,22,27,23,24);(2,4)(13,15,17,22)(14,20,21,27)(16,23,19,26)(18,24,25,28);(5,7,9,8)(6,10,11,12)(15,22)(20,27)(23,26)(24,28)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=i,e^2=hi,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi,d^c=dh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103589);