#406012

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321742
maximal normal subgroup of	#72321742

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313208);