#302018

order	512
nilpotency class	3
isoclinism	isoclinic class of #301993
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62280	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

958 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=f,d^2=ei,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=di,g^c=ghi

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, 209214);