#181010

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1890 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,f^2=i,b^a=be,c^a=ci,e^a=eh,f^b=fh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239501, #72239509
maximal normal subgroup of	#72239501, #72239509
maximal quotient of	#72239504

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88206);