#497404

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3292	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216402, #72216411, #72216412
maximal normal subgroup of	#72216402, #72216411, #72216412
maximal quotient of	D₈×#357

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404600);