#368131

order	512
nilpotency class	3
isoclinism	isoclinic class of #23892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1583 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227326, #72312202, #72312221, …
maximal normal subgroup of	#72227326, #72312202, #72312221, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275327);