#7516201

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45245	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,25)(15,27,31,21)(16,22,17,23)(18,24,20,26)(28,40,33,36)(29,43,44,37)(30,42,35,38)(32,39,34,41);(1,2)(3,4)(6,9)(10,12)(13,15)(14,21)(16,28)(17,33)(18,30)(19,31)(20,35)(22,36)(23,40)(24,38)(25,27)(26,42)(29,34)(32,44)(37,41)(39,43);(2,4)(13,16,19,17)(14,22,25,23)(15,28,31,33)(18,34,20,32)(21,36,27,40)(24,41,26,39)(29,35,44,30)(37,42,43,38);(2,4)(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,23,25,22)(15,29,31,44)(18,34,20,32)(21,37,27,43)(24,41,26,39)(28,35,33,30)(36,42,40,38);(6,9)(10,12)(13,18,19,20)(14,24,25,26)(15,30,31,35)(16,32,17,34)(21,38,27,42)(22,39,23,41)(28,44,33,29)(36,43,40,37)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7423397);