#298727

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,15)(8,18)(9,20)(10,12)(11,17)(13,19)(14,16)(21,22,25,28)(23,29,30,36)(24,32,33,27)(26,34,35,31);(1,2)(3,4)(5,7,9,17,10,8,16,19)(6,11,14,15,12,13,20,18);(2,4)(6,12)(11,13)(14,20)(15,18)(21,23,24,31,25,30,33,34)(22,26,27,29,28,35,32,36);(5,8,9,19,10,7,16,17)(6,13,14,18,12,11,20,15)(23,30)(26,35)(29,36)(31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205923);