#385884

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2535	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(5,6,7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,23)(21,25)(24,28)(26,31)(27,29)(30,32);(2,4)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,27)(21,26)(23,29)(24,30)(25,31)(28,32);(1,2,3,4)(9,12,13,14)(10,11,15,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(10,15)(11,16)(18,23)(19,26)(20,28)(22,30)(24,25)(29,31)

Polycyclic presentation:

2^9|a^2=h,c^2=eg,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293229, #72293234, #72293251
maximal normal subgroup of	#72293229, #72293234, #72293251

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293080);