#6440762

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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	#35250	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,23)(18,21)(19,22)(20,25)(24,28)(26,27);(1,2)(3,4)(6,9)(10,12)(13,15,18,22,19,21,17,14)(16,23,25,28,26,27,24,20);(2,4)(5,7,8,11)(6,10,9,12)(13,16,17,24,19,26,18,25)(14,20,21,27,22,28,15,23);(2,4);(13,17,19,18)(14,15,22,21)(16,24,26,25)(20,23,28,27)

Polycyclic presentation:

2^9|b^2=g,c^2=f,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=dh,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250181, #72256992, #72257025
maximal normal subgroup of	#72250181, #72256992, #72257025
maximal quotient of	#72308480

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347958);