#425250

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1061 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)(3,4)(5,6)(7,15,19,29)(8,12)(9,13)(10,14)(11,18,16,23)(17,26,32,33)(20,31,34,25)(21,27)(22,28)(24,30)(35,36);(5,7)(6,11)(8,20)(9,23)(10,19)(12,26)(13,29)(14,16)(15,30)(17,21)(18,24)(22,34)(25,27)(28,33)(31,36)(32,35);(5,6)(7,16,19,11)(8,12)(9,13)(10,14)(15,18,29,23)(17,31,32,25)(20,33,34,26)(21,27)(22,28)(24,30)(35,36);(2,4)(5,8,9,21,10,22,24,35)(6,12,13,27,14,28,30,36)(7,17,18,20,19,32,23,34)(11,25,15,26,16,31,29,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322946
maximal normal subgroup of	#72322946

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 332446);