#7538064

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#28452	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1905 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)(5,6,8,10)(7,11,12,9)(15,16)(20,24)(21,28)(25,26);(1,2,4,3)(7,12)(9,11)(13,14)(15,24)(16,20)(17,22)(18,23)(19,27)(21,26)(25,28);(5,7,8,12)(6,9,10,11)(13,15,18,16)(14,20,23,24)(17,26,19,25)(21,22,28,27);(6,10)(9,11)(13,16,18,15)(14,21,23,28)(17,26,19,25)(20,22,24,27);(1,3,4,2)(13,17,18,19)(14,22,23,27)(15,25,16,26)(20,21,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7445260);