#429599

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1088 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)(5,6,8,10)(7,11,12,9)(13,14,19,20,18,22,17,16)(15,21,23,27,26,28,25,24);(5,7,8,12)(6,9,10,11)(14,20)(15,23)(16,22)(17,19)(21,28)(25,26);(6,10)(9,11)(13,15)(14,21)(16,24)(17,25)(18,26)(19,23)(20,27)(22,28);(2,4)(7,12)(9,11)(13,16,19,14,18,20,17,22)(15,24,23,21,26,27,25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322424
maximal normal subgroup of	#72322424

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336795);