#196873

order	512
nilpotency class	3
isoclinism	isoclinic class of #161742
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1765 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261620, #72262032
maximal normal subgroup of	#72261620, #72262032

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104069);