#7541902

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#24162	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1831 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:

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

Polycyclic presentation:

2^9|b^2=fgi,c^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di

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, 7449098);