#6371362

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6278 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)(9,10,13,18,14,17,24,30)(11,15,20,27,21,26,33,37)(12,16,23,29,22,28,34,38)(19,25,32,36,31,35,39,40);(1,2,3,4)(9,11,13,20,14,21,24,33)(10,15,18,27,17,26,30,37)(12,19,23,32,22,31,34,39)(16,25,29,36,28,35,38,40);(6,8)(9,12)(10,16)(11,19)(13,23)(14,22)(15,25)(17,28)(18,29)(20,32)(21,31)(24,34)(26,35)(27,36)(30,38)(33,39)(37,40);(12,22)(16,28)(19,31)(23,34)(25,35)(29,38)(32,39)(36,40);(10,17)(15,26)(16,28)(18,30)(25,35)(27,37)(29,38)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314509, #72314518, #72314526, …
maximal normal subgroup of	#72314509, #72314518, #72314526, …
maximal quotient of	#72320466, #72321791

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278558);