#187827

order	512
nilpotency class	3
isoclinism	isoclinic class of #187766
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	C₂×#3508	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2518 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,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	#72305974, #72305975, #72305979
maximal normal subgroup of	#72305974, #72305975, #72305979

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95023);