#246052

order	512
nilpotency class	3
isoclinism	isoclinic class of #245999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#37673	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1638 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311631, #72311641, #72311678, …
maximal normal subgroup of	#72311631, #72311641, #72311678, …
maximal quotient of	C₂×#246052

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153248);