#7006612

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#42926	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2024 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6913808);