#187812

order	512
nilpotency class	3
isoclinism	isoclinic class of #187756
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18)(19,26,28,23)(20,24)(21,25)(22,27,31,29)(30,32);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,22)(20,29)(21,28)(23,24)(25,31)(26,32)(27,30);(17,19)(18,23)(20,27)(21,28)(22,24)(25,26)(29,30)(31,32);(2,7)(6,9)(8,15)(14,16)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302701, #72303011, #72303022
maximal normal subgroup of	#72302701, #72303011, #72303022

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95008);