#248264

order	512
nilpotency class	3
isoclinism	isoclinic class of #248263
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#35921	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214707
maximal normal subgroup of	#72214707

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155460);