#400221

order	512
nilpotency class	3
isoclinism	isoclinic class of #394464
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

901 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)(7,16,19,29)(8,13)(9,14)(10,24)(11,15)(12,20,26,17)(18,30,33,35)(21,27)(22,28)(23,34)(25,32,36,31);(7,17,19,20)(8,21)(12,16,26,29)(13,27)(18,32,33,31)(22,23)(25,35,36,30)(28,34);(5,7)(6,12)(8,18)(9,17)(10,20)(11,19)(13,25)(14,29)(15,26)(16,24)(21,33)(22,31)(23,32)(27,36)(28,35)(30,34);(2,4)(5,8,9,22,11,21,10,23)(6,13,14,28,15,27,24,34)(7,18,17,31,19,33,20,32)(12,25,29,35,26,36,16,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252649, #72256873, #72256910
maximal normal subgroup of	#72252649, #72256873, #72256910

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307417);