#7622620

order	512
nilpotency class	4
isoclinism	isoclinic class of #62636
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator	C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂×#3415	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,f^2=hi,h^2=i,b^a=bf,c^a=ci,f^a=fh,h^a=hi,f^b=fh,h^b=hi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216402, #72216441, #72216444, …
maximal normal subgroup of	#72216402, #72216441, #72216444, …
maximal quotient of	D₄×#2177, #72232006, #72232015, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529816);