#7522997

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1798 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=fi,e^2=f,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, 7430193);