#315601

order	512
nilpotency class	3
isoclinism	isoclinic class of #298603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

763 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222797);