#222853

order	512
nilpotency class	3
isoclinism	isoclinic class of #222688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	#61483	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,6,12,8,3,7,11)(4,14,16,9,5,10,15,13)(17,18,21,24,22,25,20,29)(19,26,28,23,27,32,30,31);(1,3)(2,8)(4,13)(5,9)(6,11)(7,12)(10,15)(14,16)(19,27)(23,31)(26,32)(28,30);(1,4,8,5)(2,9,3,13)(6,15,7,16)(10,12,14,11)(17,19)(18,23)(20,30)(21,28)(22,27)(24,32)(25,31)(26,29);(1,5,8,4)(2,10,3,14)(6,15,7,16)(9,12,13,11)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=h,f^2=hi,g^2=hi,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, 130049);