#285112

order	512
nilpotency class	3
isoclinism	isoclinic class of #285051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×AGL(1,ℤ/8)	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

651 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=hi,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=di,e^b=eh,g^b=gh,f^c=fi,g^c=ghi,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192308);