#182635

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257841
maximal normal subgroup of	#72257841
maximal quotient of	#72307323

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89831);