#6682786

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2662 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257274, #72257437, #72264223, …
maximal normal subgroup of	#72257274, #72257437, #72264223, …
maximal quotient of	#72264733, #72315958

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6589982);