#246007

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	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2092 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,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	#72227967, #72227985
maximal normal subgroup of	#72227967

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153203);