#392652

order	512
nilpotency class	3
isoclinism	isoclinic class of #32888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2431	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4094 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302688, #72302859, #72302866
maximal normal subgroup of	#72302688, #72302859, #72302866

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299848);