#405111

order	512
nilpotency class	3
isoclinism	isoclinic class of #393877
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	C₂×#3006	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2522 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)(7,9)(8,10)(11,12)(15,22)(16,17)(19,26)(20,24)(23,25)(27,28);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,14,18,21)(15,19,23,28)(16,20,17,24)(22,27,25,26);(2,4)(13,15)(14,19)(16,22)(17,25)(18,23)(20,27)(21,28)(24,26);(6,10)(9,12)(13,16,18,17)(14,20,21,24)(15,22,23,25)(19,27,28,26)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302575, #72302859, #72302866
maximal normal subgroup of	#72302575, #72302859, #72302866

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312307);