#363913

order	512
nilpotency class	3
isoclinism	isoclinic class of #23900
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62814	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2614 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264612, #72264922, #72264939
maximal normal subgroup of	#72264612, #72264922, #72264939

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271109);