#379819

order	512
nilpotency class	3
isoclinism	isoclinic class of #214688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,10,17,11,15,9,16)(7,14,19,31,20,12,25,27)(8,21,23,13,24,29,22,28)(18,32,33,26,34,36,35,30);(1,2)(3,4)(5,7)(6,12)(8,18)(9,19)(10,25)(11,20)(13,26)(14,15)(16,27)(17,31)(21,36)(22,33)(23,35)(24,34)(28,30)(29,32);(2,4)(5,8,11,24)(6,13,15,28)(7,18,20,34)(9,23,10,22)(12,26,14,30)(16,29,17,21)(19,35,25,33)(27,32,31,36);(5,7)(6,14)(8,18)(9,25)(10,19)(11,20)(12,15)(13,26)(16,27)(17,31)(21,32)(22,35)(23,33)(24,34)(28,30)(29,36)

Polycyclic presentation:

2^9|a^2=h,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,h^b=hi,d^c=de,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284248, #72285813, #72285823, …
maximal normal subgroup of	#72284248, #72285813, #72285823, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287015);