#370371

order	512
nilpotency class	3
isoclinism	isoclinic class of #219498
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

784 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10,14,21,16,22,15,20)(11,23,25,17,26,28,18,12)(13,29,30,19,32,35,31,36)(24,27,38,37,39,33,34,40);(2,5)(6,7)(9,11)(10,17,22,12)(13,24)(14,25)(15,18)(16,26)(19,33,36,27)(20,23,21,28)(29,37,35,40)(30,38)(31,34)(32,39);(9,12,16,17)(10,18)(11,20)(13,27,32,33)(14,28,15,23)(19,34)(21,26)(22,25)(24,29)(30,40,31,37)(35,39)(36,38);(1,3,4,8)(2,6,5,7)(9,13,14,30,16,32,15,31)(10,19,20,29,22,36,21,35)(11,24,25,38,26,39,18,34)(12,27,28,40,17,33,23,37)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226668, #72226672, #72226675
maximal normal subgroup of	#72226668, #72226672, #72226675

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277567);