#371246

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1068 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321621
maximal normal subgroup of	#72321621

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278442);