#6457373

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2290 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)(5,6,8,10)(7,9,11,12)(13,14)(15,26)(16,31)(17,23)(18,24)(19,37)(20,25)(21,29)(22,33)(27,42)(28,39)(30,40)(32,38)(34,41)(35,43)(36,44);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,29,20,30,18,28)(14,21,24,26,25,39,37,40)(16,32,33,36,34,17,27,35)(22,23,31,44,42,38,41,43);(7,11)(9,12)(13,16,19,33,20,34,18,27)(14,22,24,31,25,42,37,41)(15,17,29,35,30,32,28,36)(21,38,26,43,39,23,40,44);(7,11)(9,12)(13,17)(14,23)(15,27)(16,29)(18,36)(19,35)(20,32)(21,31)(22,40)(24,44)(25,38)(26,42)(28,34)(30,33)(37,43)(39,41);(13,18,20,19)(14,24,25,37)(15,28,30,29)(16,33,34,27)(17,35,32,36)(21,26,39,40)(22,41,42,31)(23,43,38,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=g,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364569);