#187978

order	512
nilpotency class	3
isoclinism	isoclinic class of #187944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2385 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)(19,22)(20,23)(21,24)(25,31)(26,29)(27,30)(28,32);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31);(18,24)(22,29)(23,30)(28,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^b=fi,g^b=gh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268066, #72268067, #72268070, …
maximal normal subgroup of	#72268066, #72268067, #72268070, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95174);