#425252

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
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

1070 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)(7,16,19,26)(8,13)(9,14)(10,25)(11,15)(12,17,28,20)(18,29,32,36)(21,34,33,27)(22,30)(23,35)(24,31);(5,7)(6,12)(8,21)(9,20)(10,17)(11,19)(13,29)(14,26)(15,28)(16,25)(18,22)(23,32)(24,33)(27,30)(31,36)(34,35);(7,17,19,20)(12,26,28,16)(18,21,32,33)(27,36,34,29);(2,4)(5,8,9,22,11,24,10,23)(6,13,14,30,15,31,25,35)(7,18,17,21,19,32,20,33)(12,27,16,29,28,34,26,36)

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=e,e^2=i,g^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^b=gi

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, 332448);