#393552

order	512
nilpotency class	3
isoclinism	isoclinic class of #32888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1682 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,e^b=ei,h^b=hi,g^d=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300748);