#117915

order	512
nilpotency class	3
isoclinism	isoclinic class of #117914
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,5,8,6,9,4,14)(3,10,13,15,12,16,11,7)(17,18)(19,25,27,22)(20,28)(21,23)(24,30)(26,31,29,32);(1,3,4,11,6,12,5,13)(2,7,8,10,9,15,14,16)(19,26,27,29)(22,31,25,32);(7,15)(10,16)(17,19)(18,22)(20,29)(21,27)(23,25)(24,32)(26,30)(28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286423, #72286759, #72286788
maximal normal subgroup of	#72286423, #72286759, #72286788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25111);