#378174

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

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1361 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)(5,6,10,15,11,9,17,14)(7,16,21,19,22,30,20,12)(8,13,24,29,25,23,26,28)(18,32,35,33,36,31,34,27);(1,2,4,3)(5,7,17,20,11,22,10,21)(6,12,14,30,9,19,15,16)(8,18,26,34,25,36,24,35)(13,27,28,31,23,33,29,32);(5,8,11,25)(6,13,9,23)(7,18,22,36)(10,26,17,24)(12,27,19,33)(14,29,15,28)(16,31,30,32)(20,35,21,34);(1,3,4,2)(5,9)(6,11)(7,19)(8,23)(10,14)(12,22)(13,25)(15,17)(16,20)(18,33)(21,30)(24,28)(26,29)(27,36)(31,35)(32,34)

Polycyclic presentation:

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

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, 285370);