#496014

order	512
nilpotency class	4
isoclinism	isoclinic class of #33688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

571 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2,4,3)(6,12)(7,15)(8,10)(9,14)(13,18)(16,19)(17,20);(5,6,10,9,11,14,8,12)(7,13,16,19,18,15,17,20);(5,7)(6,13)(8,17)(9,19)(10,16)(11,18)(12,20)(14,15);(1,3,4,2)(6,14)(7,16,18,17)(9,12)(13,20,15,19)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=egh,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,d^c=dh,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72221516, C₄×#33688, #72258279, …
maximal normal subgroup of	C₄×#33688, #72258279, #72258294
maximal quotient of	#72221516, #72242479, #72242485

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403210);