#196518

order	512
nilpotency class	3
isoclinism	isoclinic class of #162406
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=hi,g^2=h,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, 103714);