#410686

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2,3,4)(5,6)(7,12)(8,11)(9,10)(13,14)(15,16)(17,21)(18,23)(19,24)(20,22)(25,28)(26,27);(2,4)(6,10)(11,12)(13,15,18,25,19,26,17,22)(14,20,21,27,24,28,23,16);(5,7,9,8)(6,11,10,12)(14,21,24,23)(15,22,26,25)(16,27)(20,28);(5,8,9,7)(6,11,10,12)(13,16,17,28,19,27,18,20)(14,22,23,26,24,25,21,15)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216151
maximal normal subgroup of	#72216151
maximal quotient of	#72216072

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317882);