#360028

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1480 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279762, #72279937, #72280409, …
maximal normal subgroup of	#72279762, #72279937, #72280409, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267224);