#181607

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,3,4)(5,6)(7,15)(8,12)(9,13)(10,14)(11,16)(17,20)(18,19)(21,22)(23,32)(24,29)(25,30)(26,35)(27,31)(28,33)(34,36);(5,7,9,17,10,18,8,16)(6,11,12,19,14,20,13,15)(21,23,27,24)(22,28,31,36)(25,33,26,34)(29,35,32,30);(5,6)(7,11)(8,13)(9,12)(10,14)(15,16)(17,19)(18,20)(21,24,27,23)(22,29,31,32)(25,33,26,34)(28,35,36,30);(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,19,20,15)(21,25,27,26)(22,30,31,35)(23,33,24,34)(28,29,36,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=h,g^2=hi,b^a=be,e^a=eh,g^a=ghi,e^b=ei,d^c=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88803);