#196706

order	512
nilpotency class	3
isoclinism	isoclinic class of #161894
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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1384 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)(6,9)(7,11)(10,12)(13,14,17,21,18,22,16,20)(15,19,23,26,25,27,24,28);(5,6,7,9,8,10,11,12)(13,15,17,24,18,25,16,23)(14,19,20,26,22,27,21,28);(2,4)(13,16,18,17)(14,20,22,21)(15,23,25,24)(19,26,27,28);(6,10)(9,12)(13,16,18,17)(14,21,22,20)(15,24,25,23)(19,26,27,28)

Polycyclic presentation:

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

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, 103902);