#7612850

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,6)(3,7,4,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,28)(19,24,30,32)(20,27,23,25)(21,31,29,26);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,24,28,32)(20,29,23,21)(25,31,27,26);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,22,23)(18,25,28,27)(19,21,30,29)(24,26,32,31);(10,14)(13,16)(17,21,23,19,22,29,20,30)(18,26,25,32,28,31,27,24);(1,4,5,3)(2,7,6,8)(17,20,22,23)(18,27,28,25)(19,29,30,21)(24,26,32,31)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fi,d^2=h,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^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7520046);