#6382192

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3036	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2486 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=fi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,g^c=gi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307342, #72307351, #72307481
maximal normal subgroup of	#72307342, #72307351, #72307481

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289388);