#380995

order	512
nilpotency class	3
isoclinism	isoclinic class of #214688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,h^b=hi,d^c=de,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288191);