#195211

order	512
nilpotency class	3
isoclinism	isoclinic class of #18192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3100	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1676 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257924, #72257929, #72257939, …
maximal normal subgroup of	#72257924, #72257929, #72257939, …
maximal quotient of	#72264974, #72265643

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102407);