#186638

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×D₄×#41	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3522 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)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,19,22,18,24,17,23)(15,20,25,27,21,16,26,28);(2,4)(6,10)(7,8)(11,12)(13,15)(14,20)(16,24)(17,25)(18,21)(19,26)(22,28)(23,27);(6,11)(7,8)(10,12)(13,16)(14,21)(15,24)(17,27)(18,20)(19,28)(22,26)(23,25);(6,12)(10,11)(13,17,18,19)(14,22,24,23)(15,25,21,26)(16,27,20,28)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279227, #72309551, #72309571, …
maximal normal subgroup of	#72279227, #72309551, #72309571, …
maximal quotient of	#72279520

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93834);