#403946

order	512
nilpotency class	3
isoclinism	isoclinic class of #393877
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3460	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2860 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)(13,14,18,22)(15,23,25,28)(16,21,17,20)(19,26,27,24);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,26)(17,24)(18,25)(20,28)(21,23)(22,27);(6,9)(10,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,23)(21,28)(22,27);(2,4)(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318087, #72318124, #72318128
maximal normal subgroup of	#72318087, #72318124, #72318128

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311142);