#120155

order	512
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

622 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=di,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fh,c^b=cf,e^b=eh,f^b=fi,e^c=eh,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214111, #72216008, #72216015
maximal normal subgroup of	#72214111, #72216008, #72216015

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27351);