#187919

order	512
nilpotency class	3
isoclinism	isoclinic class of #187878
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62835	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2337 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=ghi,d^c=dh,e^c=ehi,f^c=fh,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240112, #72240120, #72240121
maximal normal subgroup of	#72240112, #72240120, #72240121

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95115);