#228329

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=eh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=deh,d^c=dhi,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343376, #72343875, #72343882
maximal normal subgroup of	#72343376, #72343875, #72343882

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 135525);