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C₂²×#1555 #6344017

order	512
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂³×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

7082 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 20 points:

(1,2);(3,4);(5,6,7,10,8,9,11,12)(13,14)(15,17)(16,18)(19,20);(6,9)(7,11)(14,17)(15,20)(18,19);(14,18)(17,19)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#18524, C₂²×#17754, C₂²×#18531, …
maximal normal subgroup of	C₂²×#18524, C₂²×#17754, C₂²×#18531, …
maximal quotient of	#72219679

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251213);