#222773

order	512
nilpotency class	3
isoclinism	isoclinic class of #222688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61087	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1059 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:

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

Polycyclic presentation:

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

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, 129969);