#249060

order	512
nilpotency class	3
isoclinism	isoclinic class of #249019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#32878	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2627 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=dhi,f^b=fh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156256);