#115999

order	512
nilpotency class	3
isoclinism	isoclinic class of #115995
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

895 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,g^2=hi,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,f^b=fh,d^c=dh,g^c=gi

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, 23195);