#119815

order	512
nilpotency class	3
isoclinism	isoclinic class of #10456
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1440
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

613 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240063, #72240064
maximal normal subgroup of	#72240063, #72240064
maximal quotient of	#72240033

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27011);