#301915

order	512
nilpotency class	3
isoclinism	isoclinic class of #301891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62245	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=fh,d^2=ehi,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,d^c=dh,g^c=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209111);