#6380856

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285505, #72285533, #72285656
maximal normal subgroup of	#72285505, #72285533, #72285656

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288052);