#108368

order	512
nilpotency class	3
isoclinism	isoclinic class of #108352
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,4,7,5,3,8,6)(9,10,13,18,14,17,25,31)(11,15,21,28,20,27,24,32)(12,16,23,29,22,30,36,40)(19,33,35,39,34,26,37,38);(9,11)(10,15,17,27)(12,19)(13,24)(14,20)(16,26,30,33)(18,32,31,28)(21,25)(22,34)(23,37)(29,39,40,38)(35,36);(1,3)(2,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,23)(14,22)(15,26)(17,30)(18,29)(20,34)(21,35)(24,37)(25,36)(27,33)(28,38)(31,40)(32,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319604, #72319773
maximal normal subgroup of	#72319604, #72319773

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15564);