#187609

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂×#3529	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2380 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)(3,4)(6,9,10,11)(13,14,19,23,18,24,17,22)(15,20,25,28,21,16,26,27);(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,24)(17,25)(18,21)(19,26)(22,28)(23,27);(2,4)(7,12)(9,11)(13,16)(14,21)(15,24)(17,27)(18,20)(19,28)(22,26)(23,25);(13,17,18,19)(14,22,24,23)(15,25,21,26)(16,27,20,28)

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209967, #72209970
maximal normal subgroup of	#72209967

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94805);