#233063

order	512
nilpotency class	3
isoclinism	isoclinic class of #233055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=dh,e^b=eh,g^b=gh,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310303, #72310308, #72310325, …
maximal normal subgroup of	#72310303, #72310308, #72310325, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140259);