#112297

order	512
nilpotency class	3
isoclinism	isoclinic class of #97710
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19493);