#7560518

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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

1996 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257238, #72257557, #72263840, …
maximal normal subgroup of	#72257238, #72257557, #72263840, …
maximal quotient of	#72264572

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7467714);