#346797

order	512
nilpotency class	3
isoclinism	isoclinic class of #346786
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2307 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253993);