#196400

order	512
nilpotency class	3
isoclinism	isoclinic class of #161458
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1532 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)(15,18,22,26)(16,23)(17,20)(19,24)(21,27,28,25);(5,6,7,9,8,10,11,12)(13,15)(14,18,20,26)(16,21)(17,22)(19,25,23,27)(24,28);(2,4);(13,16,17,24)(14,19,20,23)(15,21,22,28)(18,25,26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103596);