#7185542

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1814 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092738);