#408924

order	512
nilpotency class	3
isoclinism	isoclinic class of #394463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

824 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)(7,16,19,25)(8,13)(9,14)(10,24)(11,15)(12,17,27,20)(18,30,32,35)(21,28)(22,34)(23,29)(26,31,36,33);(1,2,4,3)(5,7)(6,12)(8,18)(9,20)(10,17)(11,19)(13,26)(14,25)(15,27)(16,24)(21,33)(22,31)(23,32)(28,35)(29,36)(30,34);(7,17,19,20)(12,25,27,16)(18,31,32,33)(26,35,36,30);(1,3,4,2)(5,8,9,21,11,23,10,22)(6,13,14,28,15,29,24,34)(7,18,20,33,19,32,17,31)(12,26,25,35,27,36,16,30)

Polycyclic presentation:

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

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, 316120);