#371182

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1258 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321621
maximal normal subgroup of	#72321621

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278378);