#481379

order	512
nilpotency class	4
isoclinism	isoclinic class of #481373
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=dh,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286888, #72287094, #72287111, …
maximal normal subgroup of	#72286888, #72287094, #72287111, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388575);