#182646

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1454 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)(5,6,9,11)(7,12,8,10)(13,14,17,19)(15,21,22,28)(16,24,25,20)(18,26,27,23);(6,10)(7,8)(11,12)(13,15)(14,18,19,27)(16,23)(17,22)(20,21,24,28)(25,26);(1,3,4,2)(6,11)(10,12)(14,19)(18,27)(20,24)(21,28);(5,7,9,8)(6,10,11,12)(14,19)(15,22)(20,24)(23,26)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=i,e^2=h,b^a=be,c^a=ch,d^a=dh,e^a=eh,g^a=ghi,d^b=dh,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257827, #72257884, #72257926, …
maximal normal subgroup of	#72257827, #72257884, #72257926, …
maximal quotient of	#72264837, #72265628

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89842);