#181408

order	512
nilpotency class	3
isoclinism	isoclinic class of #8101
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#95
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235218, #72308593, C₂×#181408
maximal normal subgroup of	#72235218, #72308593, C₂×#181408
maximal quotient of	C₂×#181408, #72309164, #72309185, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88604);