#6379070

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3485	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343372, #72343789, #72343798
maximal normal subgroup of	#72343372, #72343789, #72343798

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286266);