#497855

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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×#135	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216402, #72216413, #72216414, …
maximal normal subgroup of	#72216402, #72216413, #72216414, …
maximal quotient of	D₈×#367

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405051);