#196532

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	C₂×C₄²	C₂⁴
prime radical	C₂²×#121	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1604 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)(5,6,10,12,9,11,8,7)(13,14,17,21,18,22,16,20)(15,23,26,27,24,28,25,19);(5,7,10,11,9,12,8,6)(13,15)(14,19,22,27)(16,25)(17,26)(18,24)(20,28,21,23);(2,4)(6,11)(7,12)(15,24)(19,27)(23,28)(25,26);(5,8,9,10)(6,12,11,7)(13,16,18,17)(14,20,22,21)(15,25,24,26)(19,28,27,23)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,g^2=h,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103728);