#181592

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

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,c^a=ch,e^a=eh,g^a=ghi,e^b=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240033, #72240042, #72240051
maximal normal subgroup of	#72240033, #72240042, #72240051
maximal quotient of	#72240035

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88788);