#377935

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

979 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285131);