#373844

order	512
nilpotency class	3
isoclinism	isoclinic class of #23900
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1322 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#1868, #72342939, #72342950, …
maximal normal subgroup of	D₄×#1868, #72342939, #72342950, …
maximal quotient of	#72343093

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281040);