#377078

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#23399	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284274);