#380553

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	#34633	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1692 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)(7,10)(8,9)(11,12)(13,14,17,21,18,16,25,20)(15,19,24,27,23,22,28,26);(1,2,3,4)(6,9)(10,12)(13,15)(14,19)(16,22)(17,24)(18,23)(20,26)(21,27)(25,28);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,18)(15,22)(17,20)(19,23)(21,25)(24,26)(27,28);(7,11)(10,12)(15,23)(19,22)(24,28)(26,27)

Polycyclic presentation:

2^9|a^2=h,b^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,d^c=de,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287749);